The club later released a statement saying the abuse was "unacceptable" and that they would will not stop until the culprits were found and banned.
Some sectors of the crowd chanted "You are a monkey, Vinicius, you are a monkey," before and during Real Madrid's 2-1 win at Atletico's Metropolitano stadium on Sunday, while missiles including lighters were thrown at him and Brazilian compatriot Rodrygo as they celebrated a goal.
"I'm a big supporter of Atletico Madrid, so I was very sad," Sanchez told Politico while at the United Nations General Assembly in New York.
"I expected a strong message from the clubs against this kind of behaviour, this is what I will ask of my team. I think its important that football clubs take this kind of behaviour seriously and react."
La Liga will report the events to the Spanish Football Federation's disciplinary committee but Atletico did not comment until Tuesday afternoon.
"Atletico Madrid condemns the unacceptable chants that a minority of fans sung outside the stadium before the derby," said the club.
"A few cannot stain the image of thousands and thousands of Atletico fans who support their team with passion and respect for the opponent."
"The pain the Rojiblanco family feels for what happened is enormous. We cannot allow our fan base be related with this type of behaviour and question our values because of a minority who do not represent us.
"Our decision is firm and clear, and we will not stop until they are kicked out of the Rojiblanco family, because they cannot be a part of it."
The storm was sparked after Pedro Bravo, the head of the Spanish football agents association, said on television that Vinicius needed to stop 'acting the monkey' when dancing to celebrate goals.
Bravo apologised, saying he was using a phrase which was not intended to have racist connotations. The phrase is commonly used in Spain to mean 'playing the fool'.
Madrid mayor Jose Luis Martinez-Almeida, another Atletico supporter, has also criticised those chanting.
"They have no place, not just at a sporting event but in any sphere of life," said Martinez-Almeida.
"I think what we have to do is identify these people because I don't think they are worthy of coming in. It is absolutely reproachable and needs to be condemned.
"I say this both as the mayor of Madrid but also as an Atletico fan. I am ashamed that there are Atletico fans capable of uttering this type of racist insult."