Three second-half goals from Kwame Kizito, Thomas Abbey and Patrick Razak were enough to break down a resilient Porcupine Warriors defence.
Hearts were without their head coach Frank Elliott Nuttall, after he was given a seven-day break to visit his family in Scotland, with assistant coach Henry Wellington filling the void.
The visitors had a better possession in the opening ten minutes but Hearts soon took over. Winful Cobbinah played striker Patrick Razak through by the 11th minute but Kotoko goalkeeper Felix Annan blocked his effort and Samuel Yeboah's effort from the rebound was kept out by Awal Mohammed.
On 37th minute, Kotoko broke loose on a counter-attack and Edwin Tuffour Frimpong, who was playing under his father Frimpong Manso, curled his effort past goalkeeper Samuel Akurugu, but was ruled offside.
After the break, Kotoko midfielder Owusu Jackson had the first effort on target by the 47th minute but Akurugu pushed it to corner-kick before Obed Owusu also headed wide five minutes later.
Hearts coach Wellington brought on Kwame Kizito for Cosmos Dauda on 63rd minute and the former found the opener three minutes later. Kotoko goalkeeper Annan failed in his attempt to dribble the substitute and the striker calmly guided his shot into the empty net to the disappointment of the travelling supporters.
However, Kotoko responded two minutes later through Awal Mohammed. The centre-back headed past Akurugu from a free-kick to set the stadium ablaze but Thomas Abbey restored Hearts' lead by 81st minute, following a miscommunication by goalkeeper Annan.
Razak made it 3-1 in additional time to kill the visitors' ambition to push the game to a draw. The defeat is Kotoko's second in two months against Hearts after they succumbed to a 1-0 defeat in the Ghana Premier League matchday eight.
The return leg of the anniversary game has been scheduled to take place on May 28 at the Baba Yara Sports Stadium.