Joseph Esso, Christopher Bonney, Kojo Obeng Junior and Kofi Kordzi netted to put the Phobians through.
For their next challenge, Kim Grant's outfit will face the winners of the tie between Unistar Academy and Amidaus Professionals.
Hearts had to do it the hard way after the visitors opened the scoring in the first half. Esso brought the Phobians level on the 44th minute before Bonney converted a penalty on the hour mark to make it 2-1.
The lead only lasted for two minutes but Hearts scored again through Obeng Junior to put the score at 3-2.
Then Kordzi netted three minutes to full-time to seal the win at the Accra Sports Stadium.
Liberty Professionals join Hearts for the next round following a 1-0 victory over Skyy FC.
The Scientific Soccer Lads will square of against one of Mighty Jets and Proud United in the Round of 16.
On Wednesday, Kotoko beat Real Tamale United 3-1 to secure their ticket to the next round.
Thursday's matches were initially scheduled for Wednesday but had to be moved forward due to heavy rains.
The ultimate winners of the competition will represent Ghana in the Caf Confederation Cup next season.