The Porcupines Warriors won the game 4-1 on penalties, after the fixture, which was initially abandoned on Sunday due to heavy rains, ended goalless in regulation time in the replay.
Hearts centre-backs Vincent Atinga and Robin Gnagne missed their kicks and Sadick Adams capitalised on that to win the day for the hosts.
"It's great feeling to beat Hearts of Oak. I have to applaud the players, fans and everyone for the support. In spite of the downpour on Sunday, they came back to cheer the players on to victory which is remarkable," Polack said.
"This is the first trophy of the season for us and you can see that I am a bit emotional. I take each match serious and don't know friendly or ceremonial matches. I am happy with the win because we deserve it and believe it will motivate us in subsequent games," he added.
Kotoko will host Medeama on Sunday in matchday 21 of the Ghana Premier League and Polack insists his players will recover on time for the fixture.
"We are now focusing on the league game on Sunday. We need to maintain the high spirit for that game. A win will be good for us in our ambitions for the season," he said.
Kotoko sit fourth on the log with 32 points, eight points behind leaders Wafa.