A vicious Paulo Dybala volley and a free-kick from Miralem Pjanic put Juve in command of the tie 21 minutes in.
Carlos Bacca's hooked volley pulled one back for Milan eight minutes into the second half, before Manuel Locatelli received a second yellow card for the visitors moments later.
Milan continued to pose a threat, though, and had opportunities to force extra time, leaving Allegri far from satisfied with what he saw.
"I wouldn't say we suffered in the second half," Allegri told Rai Sport.
"We know from so many games this season that Milan never say die, no matter the result they always keep pushing and playing their football.
"We had several chances and allowed little at the back, then Dybala went off, we stopped having the extra man between the lines and that became a problem.
"At the end it was 10 against 11, it's true, but with four strikers we were all on the same line and the extra man didn't make a difference tactically.
"The extra man advantage was annulled and we annulled it. We misplaced too many passes because we went too deep.
"It's not an issue of the system so much as the attitude. Obviously the forwards have to help out in defence, but they've got good legs and quality up there, so it's entirely possible.
"A lot depends on the characteristics of the game, of the players I want to use and those at my disposal. If everyone is prepared to work hard, more quality on the field can help us.
"At the end tonight we didn't defend very well because we stopped keeping hold of the ball."