At the end Neil Lennon looked dismayed and it was easy to understand why. Celtic had played well throughout only to be undone by a deflected shot from Cristían Zapata eight minutes from time.
Sulley Muntari added a second goal for Milan four minutes later to make it even worse for the Scottish champions but it was particularly galling as Anthony Stokes had just hit the bar for the visitors before Zapata's long-range shot was deflected into the net by the Celtic defender Emilio Izaguirre.
"I am bitterly disappointed because for long periods of the game we were the better team," said Lennon. "To come away from the San Siro [disappointed] not to have picked up a point or three points is a mark of how well we have developed and how well we played tonight.
"We are here to qualify and nothing changes my view after that performance. The scoreline flattered Milan. I don't think there was two goals between the teams.
"I thought we were the better team playing the better football. We had really good chances tonight and that will be one regret going away from here."
Milan started brightly and after only five minutes Fraser Forster was forced to parry a snatched volley from Mario Balotelli.
On 20 Celtic were reacquainted with the man whose two goals for Juventus knocked them out of last season's competition in February, when Alessandro Matri's twisting header from a right-wing cross obliged Forster to turn it round a post for a corner. While defensively sturdy, Celtic's only real attack in the first half-hour came courtesy of the German referee Wolfgang Stark's award of an indirect free-kick within the Milan penalty area after Abbiati was ruled to have handled incorrectly after collecting a deflection from one of his defenders.
The cropped red hair of the anxious Lennon became a permanent feature on the touchline as his side failed to make inroads against a Milan defence lacking both the injured Daniele Bonera and Ignazio Abate. Also watching from the stands was Kaká, whose spectacular individual strike in extra-time had settled things between the clubs in 2007 when Lennon himself was still starring on the other side of the white line.
A low drive from Stokes, who neatly cut inside Zapata on the left shortly before the interval, provoked Lennon into kicking up turf in frustration in front of the Celtic dugout. But he would have been pleased to see his side go in at half-time still goalless after a largely anonymous Balotelli had a deflected shot tipped round a post.
After the break Muntari missed a great chance while Celtic's in-form striker Stokes hit the bar with that free-kick. It was downhill from there, though, as Milan broke the deadlock with their next attack. Within two minutes Forster was beaten again despite making a great save from Balotelli's free-kick, as Muntari knocked in the rebound.
The Milan manager, Massimiliano Allegri, was impressed by the visitors. "We had luck a couple of times in the game tonight but it was an important victory for us," he said. "Celtic are a very good team. I believe they still have chances of qualifying out of the group."