The Phobians recorded a 2-0 win over Danbort FC to advance to the Round of 32.
The Accra giants have now emerged as huge favourites after rivals Asante Kotoko crashed out of of the competition.
Two-time Ghana Premier League champions Aduana Stars have also bundled out of the competition after losing 2-1 at home to Bechem United.
Berekum Chelsea, who clinched the Ghana Premier League in 2012, have also exited the competition.
The situation has left Hearts as favourites despite the threat of two-time FA Cup holders Medeama.
Hearts are the most successful club in the FA Cup after winning it 10-times.