The home side was struggling against
However, it got a huge slice of luck when defender
The Hoops goalkeeper was beaten again four minutes later despite making a great save from Mario Balotelli's free-kick, as midfielder Sulley Muntari knocked in the rebound.
Much was made of AC Milan head coach
Much of that, however, was down to the performance of Celtic.
Forster had to make several saves early on and one near the end of the first half, but in between, the visitor enjoyed more than its fair share of possession.
There was a confidence in the way the Scottish champion kept the ball albeit it without threatening Milan keeper
After the break Muntari missed a great chance while Celtic's in-form striker
The two late goals, though, were a sickener for Celtic which takes on Barcelona in its next group fixture at Parkhead at the start of October with Ajax also waiting in the wings.
However, on its overall display, it can go forward in the group with some confidence.
Allegri complained on Tuesday of a long list absentees including Kaka and Montolivo but he could still afford to leave Robinho on the bench as he chose former Manchester City forward Balotelli and
Celtic midfielder Joe Ledley dropped out with a groin injury to be replaced by
Milan, a seven-time winner of European club football's top trophy, took control of the game after
Forster parried Balotelli's powerful volley clear four minutes later, with midfielder
Moments later, right-back
Forster then had to concede a corner with another decent save from Matri's flashing header after
Just after the half-hour mark Nocerino sent a dipping volley just over the bar but five minutes from the interval a careless pass towards his own goal from wide on the right from Milan midfielder
Seconds into added time in the first half, Forster had to scramble down low to his left to save a low drive from Balotelli which deflected off Hoops defender
The visitor started the second half brimming with belief and came close five minutes after the restart when Samaras drove just past the post from 25 yards.
The Milan fans were becoming increasingly frustrated as Celtic encroached into the box with menace although the visitor escaped again in the 62nd minute when Muntari missed a sitter, heading Zapata's cross over from three yards.
Celtic had its best chance in the 67th minute when Commons played in skipper
In the 79th minute Celtic substitute
More woe visited Celtic when Brown was booked for a foul on Balotelli, whose curling free-kick from 25 yards was brilliantly saved by Forster, only for Muntari to bundle the rebound in.