Goals from Kaka, Cristian Zapata and Mario Balotelli were enough to give Milan all three points and coupled with Ajax’s win over Barcelona it means the Scottish champions go out of Europe before Christmas.
Neil Lennon’s side went into the game knowing it was a must-win encounter if they were to keep alive their Champions League hopes, but they were easily swept aside by Milan who put their domestic problems behind them to boost their chances of reaching the knockout stages.
Celtic had a sight of goal after just three minutes when Derk Boerrigter got on the end of Charlie Mulgrew’s inswinging corner, but the Dutchman failed to connect with the ball properly and the chance went begging.
Balotelli had a half chance for Milan seconds later when he got the better of Efe Ambrose only to drag his shot from the edge of the box wide of the target.
Celtic were looking to get off to a flying start and they had a chance to break the deadlock on nine minutes when the ball broke to Beram Kayal inside the box, but under pressure from Zapata he failed to test Christian Abbiati in the Milan goal.
Milan hit the front three minutes later with Celtic gifting the Italians an easy goal with some shoddy defending. Kaka found himself unmarked under six yards out from Valter Birsa’s corner and the Brazilian duly dispatched his header past Fraser Forster.
Celtic were struggling to make any inroads into the Milan defence and the visitors looked dangerous on the break with Balotelli fizzing a shot just over the crossbar on 25 minutes after some fine approach play by Kaka.
Kaka, who was rolling back the years with an impressive performance, was inches from doubling the lead on 29 minutes when he saw his curling effort from the edge of the box fly just wide of the target.
Charlie Mulgrew had a decent opening for Celtic on 39 minutes when the ball fell to him inside the box, but his shot lacked the power to trouble Abbiati.
Celtic really should have drawn level two minutes after the restart when Mulgrew, after seeing his initial free-kick blocked, lifted the ball into the box for the unmarked Virgil van Dijk who volleyed his effort straight at Abbiati with the goal at his mercy.
The home side were made to pay for that miss on 49 minutes when a corner found substitute Antonio Nocerino free in the box and he played the ball back across the face of goal for Zapata to tap home at the far post.
Celtic thought they had pulled a goal back two minutes later when van Dijk headed home Joe Ledley’s cross, but his celebrations were cut short by an offside flag.
Milan put the game beyond Celtic just before the hour mark when Riccardo Montolivo sent Balotelli clear on goal with a superb through ball and the former Manchester City striker held off Ambrose before slotting past Forster.
James Forrest had a chance to pull a goal back on 62 minutes when he got on the end of Mikael Lustig’s pass, but the winger could only stab his effort wide of the target to sum up Celtic’s night.
Kaka was replaced 10 minutes from time with the Celtic fans showing a nice touch by applauding the Brazilian off the pitch as they saw their European hopes ended on a disappointing night in Glasgow.