The early game in Italy's Serie A offered Gennaro Gattuso's men the chance to leapfrog Lazio into fourth place - the last slot for qualification to next season's Uefa Champions League.
But they fluffed their lines at a team who had only collected nine points from the 17 games before Milan's visit.
With Jose Mourinho newly roaming since his dismissal from Manchester United on 18 December, Gattuso will be under increasing scrutiny and mounting pressure to arrest the slump that has brought only one win in seven games.
There is speculation that his goal shy striker Gonzalo Higuain could move during the January transfer window.
"No striker is happy when he does not score," said Gattuso. "But it's also hard for the whole team. Higuain is going through a difficult moment, like the team and as long as he is here he remains an important player to be exploited to the full," Gattuso added.
Higuain scored 16 goals last season for Juventus before joining Milan on loan. The 31-year-old has netted five this term. But he has not hit the target since 28 October.
The Argentina international's old club, however, are prospering in his absence. With 100 million signing Cristiano Ronaldo, Juventus are top of the league with 49 points from 17 games.