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It all started so well for Celtic, racing into a 2-0 lead at the San Siro, but that's a fourth defeat in five group matches for Neil Lennon's men.
The visitors had a decent spell in the second half but were undone by more hesitant defending.
AC Milan winger Jens Petter Hauge was a real handful, scoring the third goal and setting up the fourth.
AC Milan 4-2 Celtic
Jeremie Frimpong hasn't stopped running all night and the Celtic teenager lets fly with a fierce, angled shot. Gianluigi Donnarumma gets down well to push it away.
AC Milan 4-2 Celtic
The game is just fizzling out now as we hit 90 minutes, with three more added.
AC Milan 4-2 Celtic
Ryan Christie with another free-kick attempt. This one trundles well wide.
The midfielder is then replaced by striker Patryk Klimala.
AC Milan 4-2 Celtic
Pat Bonner
Former Celtic goalkeeper on BBC Sportsound
Quote MessageAgain it's awful defending from Celtic. The first job of a full back is to get tight and stop the winger from dictating things. Elhamed is just backing off and backing off.
Brahim Diaz
Celtic just can't handle Jens Petter Hauge.
Defenders stand in a daze as the Norwegian jinks his way into the penalty area and then pokes a pass through to Brahim Diaz, who does well to dink a close-range finish past Vasilis Barkas.
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AC Milan 3-2 Celtic
Celtic captain Scott Brown trots off and we get a rare glimpse of Ismaila Soro.
Rangers v Standard Liege (20:00)
Rangers: McGregor, Tavernier, Goldson, Balogun, Barisic, Arfield, Davis, Kamara, Roofe, Kent, Morelos
Substitutes: McLaughlin, Bassey, Hagi, Itten, Zungu, Patterson, Aribo, Barker, Stewart, Barjonas, Dickson, King
Standard Liege: Bodart, Fai, Dussenne, Laifis, Jans, Bope Bokadi, Shamir, Bastien, Lestienne, Cop, Tapsoba
Substitutes: Gillet, Henkinet, Sissako, Sylvestre, Gavory, Avenatti, Raskin, Carcela, Balikwisha, Pavlovic, Landu, Siquet
AC Milan 3-2 Celtic
Pat Bonner
Former Celtic goalkeeper on BBC Sportsound
Quote MessageThis second half has been much better from Celtic. They've kept the ball well and that can help give your defence a break. They've controlled it for most of this half.
AC Milan 3-2 Celtic
Celtic skipper Scott Brown goes into the referee's book for halting a counter-attack with a clip on the heel of Theo Hernandez.
AC Milan 3-2 Celtic
Ryan Christie lines up a free-kick 25 yards out and it looks like the ball is curling into the top corner, only for Gianluigi Donnarumma to make a fine save, using every inch of of his giant frame to push it away.
AC Milan 3-2 Celtic
Defender Kristoffer Ajer joins the Celtic attack, ghosting past a couple of opponents to get into the box. He tries to find Odsonne Edouard with a low pass but Alessio Romagnoli gets in the way again.
AC Milan 3-2 Celtic
Tom Rogic is replaced by Olivier Ntcham after picking up a booking for a frustrated challenge.
AC Milan 3-2 Celtic
Celtic enjoy a good spell of possession and Callum McGregor almost gets on the end of a smart pass from Ryan Christie but Alessio Romagnoli just gets there first to mop up.
AC Milan 3-2 Celtic
Pat Bonner
Former Celtic goalkeeper on BBC Sportsound
Quote MessageHow many times have Milan given the ball away in a dangerous area? That's where Celtic have had their chances, from Milan's poor passing.
AC Milan 3-2 Celtic
Here comes Jeremie Frimpong galloping into the home penalty area and he does well to chop inside to create room for a shot, but the Celtic full-back drags his left-foot effort just wide of the near post.
AC Milan 3-2 Celtic
AC Milan make two more changes as Brahim Diaz and Ismael Bennacer come on.
Hakan Calhanoglu and Franck Kessie are the men making way.
AC Milan 3-2 Celtic
Pat Bonner
Former Celtic goalkeeper on BBC Sportsound
Quote MessageTonali has come on and is using his experience to control the game for Milan. I just can't see Milan throwing away this lead like Celtic did in the first half
AC Milan 3-2 Celtic
That's been five whole minutes without a defensive lapse from either side.
How long can they keep this up?