Manchester United made it eight goals in two games with their 4-0 thrashing of Swansea.
Newcomers Huddersfield and West Brom also made it two wins from two, while champions Chelsea got off the mark with a thrilling win over Tottenham at Wembley.
Bournemouth, Newcastle, Brighton, Crystal Palace and West Ham are still looking for their first points of the season.
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Goalkeeper - Jack Butland (Stoke) Jack Butland made six saves against Arsenal - the joint second most in a Premier League game this season
It's great to see Jack Butland between the sticks again and his performance was outstanding against an Arsenal side who were denied two penalties and a perfectly good goal.
I have no intention of being an apologist for Arsenal - they have caused me enough heartache over the years - but how they never came away with at least a point against Stoke I will never know. Gunners duo Hector Bellerin and Danny Welbeck had perfectly clear penalties turned down, and how anyone could argue Alexandre Lacazette's toe was offside is so tenuous it beggars belief.
What's the game coming to? Is this what TV evidence was created for: to examine if a toe is in an offside position or not? What next - forelocks?
What TV evidence did clearly portray was performance - and that's what it was designed to do. In that regard, Butland is slowly coming back to his best and, in my opinion, is currently England's number one choice.
Of course, he will have to prove that by having an outstanding, injury-free season - but if he achieves that it will be great news for Stoke and even better news for England.
Defender - Andrew Robertson (Liverpool) Andrew Robertson made three key passes against Crystal Palace - more than any other Liverpool player
Now this lad looks promising. It's been a long time since Liverpool have had a full-back with a left foot like Andrew Robertson.
His crosses, against a Crystal Palace side much better organised and purposeful than they were last week against Huddersfield, reminded me of Germany and Hamburg legend Manny Kaltz.
Although Kaltz was a right-back, his quality of crossing was legendary and Robertson appears to have that ability. The young Scot also looks comfortable on the ball with an eye for goal - a feature clearly demonstrated when he forced Palace keeper Wayne Hennessey to tip the ball over the bar at full stretch.
The issue is, of course, can the full-back defend? Palace didn't really test him, but better teams will.
Robertson is obviously part of Liverpool's future, unlike midfielder Philippe Coutinho. There is no doubt that Coutinho has already mentally left Anfield and the club would be foolish not to accept his position.
However, what the Reds don't have to accept is any old Barcelona offer. They are perfectly entitled to squeeze every last penny out of this deal, and when they do finally settle on the transaction - and they will - I hope it makes Barcelona's eyes water.
Defender - Eric Bailly (Manchester United) Eric Bailly scored his first Premier League goal for Manchester United and made eight clearances, with only Phil Jones making more for the Red Devils
I want to like this player, but he always gives me the 'willies'.
Eric Bailly, on his day, is a class act - a point clearly demonstrated against Swansea - but why does he give me a sense of nervous apprehension every time he plays?
I can't help feeling he's a tackle away from a red card or giving a penalty away. Against the likes of Swansea he will be fine, but it's in the big games, when it really matters, that I find myself asking the question: can he be trusted?
Defender - Harry Maguire (Leicester) Harry Maguire has now scored a goal and assisted another in his first two Premier League appearances for Leicester - he also made more clearances than any other Foxes player against Brighton (five)
This was almost a complete performance from Harry Maguire against a Brighton team in desperate need of quality.
I saw Maguire play for Hull against Chelsea last season at Stamford Bridge and he was outstanding. Since then, he has developed into a Premier League player and looks like making the transition from Hull to Leicester City quite comfortably.
During that time he has won admiring comments from pundits like Martin Keown and Gary Lineker. Having worked with both, I can assure you that neither of my BBC colleagues are easily pleased, particularly when it comes to evaluating defenders, with the former an uncompromising centre-back and the latter an exceptional centre-forward. I get the feeling that both would have welcomed the no-nonsense Maguire in their teams.
However, Brighton failed to put Maguire under any pressure whatsoever - a feature captured when the camera caught owner Tony Bloom with his head in his hands in frustration, having seen another one of his team's attacks come to nothing.
There is one simple solution to all this anguish, Mr Bloom: put your hand in your pocket and bring in the quality the club requires to survive in the best league in the world. Otherwise there will be far more moments like this.
Defender - Antonio Valencia (Manchester United) Antonio Valencia gained possession more often than any other player on the pitch in United's 4-0 win at Swansea (12 times)
What an excellent choice of captain by United manager Jose Mourinho. Antonio Valencia proved to be an excellent player under former boss Sir Alex Ferguson and looks even more valuable to United in this season's title-chasing campaign under Mourinho.
The Ecuadorian can still deliver the first-time cross when in advanced areas - unlike most full-backs in the Premier League - and is as quick in recovery as any defender in the top flight.
Against a totally outclassed Swansea, Valencia looked unflappable and reflects a calm demeanour not normally associated with a Mourinho side.
The Swans, meanwhile, hardly laid a glove on United (regardless of the score on 79 minutes) and with midfielder Gylfi Sigurdsson gone to Everton, the measured manager Paul Clement needs to replace his best player with another talisman very quickly, otherwise his season could be over quicker than a verse of Land Of My Fathers.
Midfielder - Aaron Mooy (Huddersfield) Aaron Mooy became the 16th Australian to score a Premier League goal by netting the winner against Newcastle
Last week it was striker Steve Mounie who stole the show. This Sunday against Newcastle - another promoted team short in quality - it was Aaron Mooy who ripped the heart out of the Magpies with a moment of sheer magic.
I said last week, after the performance against Crystal Palace, that Huddersfield were going to have to call on players like Mooy and Mounie if they intend to stay in the league. His goal was of the highest quality, but more importantly the Australian possesses the tools to unlock any defence.
Meanwhile, Newcastle have one of the best managers in the world who is having to suffer the indignation of an owner who cannot match his ambition. So can someone explain to me what's Rafael Benitez doing there?
Midfielder - Henrikh Mkhitaryan (Manchester United) Henrikh Mkhitaryan has already registered four Premier League assists - two more than any other player this season. He created two goals in the 4-0 win over Swansea.
To keep midfielder Ander Herrera out of the team tells its own story, but Henrikh Mkhitaryan is doing much more than that at the moment. The Armenia international complements this new-look Manchester United team perfectly.
This is a very uncomplicated footballer. He doesn't fuss when things don't go his own way and loves to get involved in the action.
Mkhitaryan was responsible for striker Romelu Lukaku and midfielder Paul Pogba's goals, which both came within a matter of minutes. It was also Mkhitaryan's performance that put the game well out of Swansea's reach.
Midfielder - Paul Pogba (Manchester United) Paul Pogba has now scored in four of his past five Manchester United games in all competitions. Against Swansea, the Frenchman also assisted Anthony Martial's fourth goal.
Pogba's header from the corner was impressive during the 4-0 win over Swansea, but why on earth did defender Federico Fernandez not just clear the rebound from the underside of the bar instead of trying to protect his goalkeeper? Lukasz Fabianski didn't need protecting, he needed a clearance.
To be perfectly honest, it wouldn't have mattered anyway because, without Sigurdsson, the chances of Swansea winning this fixture were so remote it wasn't even worth a bet.
Regardless, this was a Pogba performance worth noting. Apart from the ridiculous brown patch in his hair, he produced a game free of the showboating and scored a goal of real class. His yellow card was justified, but what took place thereafter was so measured and so utterly professional and selfless, it was in stark contrast to anything I had seen from the player before.
I keep hearing pundits (mostly his mates) telling me Pogba is capable of almost anything. Well, I think we are finally starting to see it. It is about time.
Midfielder - Marcos Alonso (Chelsea) Marcos Alonso scored his second Premier League double for Chelsea and no Blues player has scored more away from home since the start of last season
All credit to Blues manager Antonio Conte for the way he reconstructed a side ravaged by injury and suspension to face Tottenham and beat them at Wembley. Premier League fixtures don't come much bigger than this and managers need players with a strong mentality and bags of courage.
Up steps Marcos Alonso. The wing-back could have easily ducked the moment and let Willian - also a dead-ball specialist - take the free-kick. But he didn't, he took control of the situation and buried it. The Spaniard didn't stop there either - his overall contribution to this Chelsea performance was immense. How this lad hasn't yet emulated his father and grandfather and represented Spain is a mystery to me.
This victory will also solidify Conte's managerial position. Make no mistake, the Italian has not handled the Diego Costa situation well and his top scorer last season will be very hard to replace. But the way the Chelsea players have rallied behind their boss to win such an important game speaks volumes about the manager and his players.
Forward - Javier Hernandez (West Ham) Javier Hernandez first Premier League goals for West Ham came from both of his shots on target against Southampton
West Ham's day fell apart when, after missing two glorious chances, midfielder Marko Arnautovic threw an elbow into the head of Southampton's Jack Stephens.
His sending off was totally justified and Hammers boss Slaven Bilic is now under the sort of pressure he brought Arnautovic in to alleviate after just two games.
Thank goodness Bilic has invested in the services of someone who is considerably more reliable and far less combustible. Javier Hernandez scored two goals of the type that has become his hallmark during his career.
I have no doubt the Hammers are going to be all right this season, but I'm less sure about Bilic's future after their mauling at Old Trafford last week and now this spirited defeat with 10 men at St Mary's.
However, if Bilic can hold on until striker Andy Carroll is fit, he could have a really exciting forward line, with the Englishman, Hernandez and a slightly reformed Arnautovic as a front three.
Forward - Jese Rodriguez (Stoke) Jese Rodriguez scored on his Premier League debut for Stoke and also made three key passes, more than any other Potters player
Once Neymar arrived at Paris-St Germain, Jese Rodriguez was out of that club like a shot.
Stoke fans had every right to be stunned by the arrival of the former Real Madrid forward. Not since the arrival of Switzerland winger Xherdan Shaqiri have Stoke fans seen such a high-profile player land on their doorstep practically unannounced.
It clearly suited Jese to come to the Potteries - it is Premier League football after all, and not even Neymar is getting that.
Manager Mark Hughes is almost beside himself having landed the player and couldn't resist putting the Spaniard straight into the team to face Arsenal after only a few days of training with the squad. What a stroke of good management.
Jese needs somewhere to play, not to mention an environment where he will be loved. The Stoke fans will bring the best out of him - all he has to do is acknowledge it and let them know he cares.
Source: bbc.com