Carlos Tevez gave Juve a 14th-minute lead in Turin, racing through on goal and producing a tidy finish under Diego Lopez, even though replays suggested he was marginally offside.
Milan hit back just before the half-hour mark, though, as debutant Luca Antonelli met Alessio Cerci's corner with a glancing header.
But Juve eventually cruised to a comfortable win thanks to further goals from Bonucci and Alvaro Morata, and the centre-back was pleased with the team's display of determination.
He told Sky Italia: "We hadn't risked anything and then conceded a goal that we really could have avoided, even from just not giving away the corner.
"[But] this is Juventus; we fight back when it is needed and we brought out all of our determination to win.
"There is a long road ahead of us. We need to be focused and hungry for success in every game in Serie A, the Champions League and Coppa Italia."
Claudio Marchisio was not quite so happy with Juve, though.
"It wasn't an easy game," the midfielder added. "It is true that we managed to get right back in front after their equaliser, but in the second half we did not come out with the right attitude and we were under pressure.
"It wasn't a proper Juventus performance. After the third goal we controlled the match again, but at the moment we are struggling to keep our concentration for the full 90 minutes."
‘He instantly popped into our ratings as one of the best U21 midfielders in Europe. The impressive thing is Slot has made him into more of a no.6' Inside Ryan Gravenberch's transformation at Liverpool
‘I don’t think Liverpool would look at Ollie Watkins, a striker isn’t a pressing issue for them – it’s Arsenal who need one’ Former Reds star explains why his old club don’t need an out-and-out forward this summer
