Clarence Seedorf's charges applied themselves well in the first half at San Siro and saw the likes of Kaka, Montolivo and Andrea Poli come close to breaking the deadlock.
However, Juventus dealt a hammer blow to the hosts shortly before half-time when Fernando Llorente tapped home, and the visitors cemented their position on 68 minutes with a stunning strike from Carlos Tevez that hit the underside of the crossbar on its way in.
Midfielder Montolivo felt that Milan were unfortunate to lose the game, but believes that the players can take pride in the way they acquitted themselves against the defending champions.
"It's disappointing to talk about the loss after such a good performance," he told the club's official website.
"We deserved to take the lead in the first half and we went in at half-time a goal down. It's very tough but we have to keep on this road that we're on with this attitude and this desire to play the game.
"We need continuity and that's what we're looking for. If we don't win then it's obvious we have to improve.
"I think that in the big games, the performances have always been there. We have to improve in certain situations where Juve punished us but today we showed good intensity and balance.
"We dropped a bit in the second half but it's tough to talk about a loss when we deserved a different result."
The setback leaves Milan in 10th position in Serie A with 35 points, some 17 adrift of a UEFA Champions League spot.
