Two goals in the final two minutes robbed Celtic of their first away Champions League point.
Filippo Inzaghi touched in a low cross from Manuel Rui Costa in the 89th minute and, a minute later, a deflected free-kick from Andrea Pirlo rubbed salt in the wound.
Until then, it looked as if a 74th minute goal from Stanislav Varga - cancelling out a strike from Andriy Shevchenko - would be enough to give Celtic a point.
Milan made their intentions clear from the kick-off and there was a lucky escape for Celtic after barely four minutes.
Joos Valgaeren was caught loitering on the edge of his area and Kaka swooped to fire in a shot which David Marshall did well to turn aside for a corner.
It was relentless early pressure from Milan and they took their reward in the ninth minute.
Kaka threaded the ball through for Shevchenko and he clipped the ball beyond Marshall with the aid of a deflection off Valgaeren to mark his 28th birthday.
Celtic settled after this but rarely had a look at Dida's goal. Varga sent a header wide from a Stilian Petrov corner midway through the first half and Henri Camara had a couple of chances before the break.
He swiped a shot wildly over after John Hartson had set him up and, on the stroke of the interval, he failed to get any contact on a carefully weighted cross from Chris Sutton.
But Celtic were fortunate just to be a goal behind by that stage. A poor clearance from Bobo Balde fell to Cafu but he failed to get any power behind his shot and Marshall gathered easily.
Balde then did well to get behind his goalkeeper and block an effort from Shevchenko after Jon Dahl Tomasson had created the opportunity.
There was a major let-off a minute from half-time when Balde was caught in possession by Shevchenko who fed Kaka but the Brazilian was denied by an alert Marshall.
Camara forced a save from Dida at the start of the second half after Hartson had headed the ball on as Celtic finally mounted something of some substance.
Celtic were much improved in the second half but lost Alan Thompson with a hamstring injury in the 59th minute with Juninho replacing him.
Dida had to race from his line to prevent Camara having a clean strike on goal as Milan looked increasingly uncertain.
Camara had another opportunity in the 71st minute when Hartson played him in but he could not get any power in his effort and Dida was untroubled.
Celtic won a series of corners and Varga cashed in on one three minutes later when he powered in a header from a Petrov corner.
But then came the late show from Milan as they stole all three points.