Daniel Afriyie scored the opener for the Phobians in the 50th minute before the attacker assisted Emmanuel Nettey for the second goal.
The visitors Windy Professionals reduced the deficit later on in the game.
However, goals from Benjamin Afutu and Umar Manaf ensured that Hearts of Oak showed their superiority over the lower tier side.
The Phobians would know their next opponents in the next draw scheduled to be held in the course of the week.
Hearts of Oak have won the FA Cup 10 times which more than any club in Ghana.
Watch highlights of the match below
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