Celtic were backed by the cacophony of noise synonymous with their Champions League nights at home on Tuesday, but slack set-piece defending decisively undermined their efforts in this must-win Group H encounter.
Kaka and Cristian Zapata were the beneficiaries either side of half-time, leaving Mario Balotelli to seal the points on a night when the enigmatic Italy international proved a thorn in the side of the home defence.
Ajax's shock win over Barcelona means Celtic are unable to even secure the consolation of a Europa League place, while Milan must avoid defeat against the Dutch champions at San Siro in a fortnight's time to secure their passage.
|By Chris Myson
That was not the night Celtic had planned. Coming into the game with everything to play for, all of a sudden Neil Lennon's side can kiss goodbye to European football for another season.
The Scottish champions will be frustrated by the manner of the first two goals they conceded - both from corners - but ultimately they have not been the same side in the Champions League this season.
For AC Milan, a controlled and impressive performance eases the pressure on Massimilano Allegri and it was great to see Kaka showing flashes of his old brilliance again.
Balotelli led Milan's attack, while Valter Birsa and Riccardo Montolivo came in for Alessandro Matri and Sulley Muntari after Saturday's 1-1 draw with Genoa.
Derk Boerrigter should have put Celtic ahead on his first Champions League start for the club after just two minutes, but he scuffed a back-post volley after Milan goalkeeper Christian Abbiati made the first of a series of misjudgements from Charlie Mulgrew's corner.
A minute later, Balotelli botched a chance to open the scoring as he dragged wide having escaped the attentions of the home defence.
Milan defender Zapata produced a vital last-ditch tackle when Mulgrew's ninth-minute shot fell to Beram Kayal in front of goal, and the visitors were ahead soon after.
Kaka was left completely unmarked to nod home at close quarters as goalkeeper Fraser Forster elected to stay on his line from Birsa's right-wing corner.
Daniele Bonera then made a timely block after Mulgrew picked out Boerrigter, but Celtic were almost hit on the break immediately as Kaka released Balotelli to drill narrowly over.
Celtic continued to press manfully before the break but their fate was sealed by two key incidents shortly after the resumption.
Centre-back Virgil van Dijk smashed a volley into Abbiati from point-black range when it might have been easier to score in the 47th minute - and his defensive colleagues were once again undone from a set-piece two minutes later.
Birsa's delivery was superb again, and half-time substitute Antonio Nocerino knocked across goal for Zapata to apply the finish.
To compound Celtic's misery, Balotelli sealed the points with an hour played after Van Dijk had headed home from an offside position.
The former Manchester City striker demonstrated his pace, power and poise to hold off Efe Ambrose and finish Montolivo's lofted pass.
Forrest almost reduced the arrears when he poked wide from Mikael Lustig's ball with the impressive Nigel de Jong in close attendance.
The game petered out as Celtic's enthusiasm faded and Massimiliano Allegri's men kept their hopes of reaching the last 16 very much alive.