The Old Lady beat the Coppa Italia holders 4-2 after extra time, and the centre-back was pleased with his side's performance in what was a 'challenging' game for him due to the recent controversy over the Scommessopoli scandal, which eventually saw his name cleared.
"We have to make sure this [Supercoppa win] is just the beginning. We have a long and demanding season ahead," Bonucci was quoted as saying on the club's official website.
"Mentally the game was challenging for two reasons. Because I had all that is gone on away from the pitch behind me, and also because of the dangerous opponents on the pitch.
"Napoli waited for us and we perhaps gave them too much space. We came good though. But that's Juve for you, a team that proves they are better than the rest on the pitch."
Juventus get their Serie A defence started against Parma in two weeks' time.