The Hoops fell behind early on though after being caught cold by a Milan corner, with Kaka nodding home from close range.
Milan pressed home their advantage from another set piece early in the second half as Cristian Zapata scored at the far post, before Mario Balotelli put the game beyond doubt soon after.
Look back at an impressive display from Massimiliano Allegri's men with Sports Mole's minute-by-minute commentary below.
6.48pmEvening European football fans. Thanks for joining Sports Mole for a Champions League clash between Celtic and AC Milan, with second place in Group H still very much up for grabs as we edge closer to finding out who will reach this season's last 16.
6.51pmCeltic are swimming in the last-chance saloon as far as qualification goes tonight, after winning just one of their first four games in a fiercely competitive Group H. Neil Lennon's men know that victory this evening is an absolute necessity if they want to match their impressive run to last season's knockout stages, hoping home advantage will make the difference for them once again.
6.54pmThe Bhoys are currently bottom of Group H, but would leapfrog Milan with a priceless win tonight. Milan meanwhile could all but secure their place in the knockout stages with a positive result this evening, although they travel to Glasgow on the back of their worst start to a Serie A campaign since 1981.
6.57pmMilan boss Massimiliano Allegri has managed to win just three of Milan's 13 Serie A matches this season, leaving them languishing in mid-table and putting his position under intense scrutiny. The Rossoneri's Champions League campaign has offered a welcome distraction though, with victories two draws and a victory putting them in pole position to finish behind Barcelona in a qualification spot as we approach the business end of the group stages.
7.00pmThe Rossoneri's 2-0 win over Celtic when the two sides last met in September barely tells the story of a clash the Hoops largely dominated, before being caught cold by two goals in the final 10 minutes. Giorgios Samaras had struck the crossbar just moments before that late collapse, giving Celtic even further incentive to exact revenge tonight.
7.03pmLennon has a pair of key midfielders missing through suspension tonight, with Nir Briton and Scott Brown both banned. James Forrest is fit and back in contention for a starting spot, with Samaras likely to lead the line in attack.
7.06pmMilan meanwhile have had their poor start to the campaign has been compounded by a growing injury list, with the likes of Stephan El Shaarawy and Mattia De Sciglio still on the treatment table. Sully Muntari and Philippe Mexes are also almost certain to miss out tonight, putting experienced figures such as Kaka under even more pressure to perform in their absence.
7.09pmBREAKING TEAM NEWS: We have just received word on the starting XIs from Celtic Park, with Lennon deciding that Forrest is fit to start out wide. A full look at both lineups will follow in just a few moments.
7.16pmPlenty of changes for Celtic tonight then, with Samaras restored to the centre of attack in place of Teemu Pukki. Mikael Lustig is back at full-back, while Biram Kayas comes in to compensate for Brown's absence in central midfield.
7.19pmMilan meanwhile have dropped Alessandro Matri in attack, with Valter Birsa coming in to play alongside Kaka in an advanced midfield role. Mario Balotelli will lead the line despite missing his second penalty of the season against Genoa last weekend, while there is no place for Robinho in the starting XI.
7.22pmDespite Muntari's absence through injury Milan still have plenty to experience in central midfield, with Riccardo Montolivo returning to play alongside Nigel de Jong and Andrea Poli. The Rossoneri will be confident of winning the midfield battle, but a makeshift centre-back partnership including Daniele Bonera will give visiting fans cause for concern.
7.25pmLennon has readily admitted that tonight's clash is win or bust, so Celtic to play with the kind of intensity and atmosphere that has made Celtic park an imposing place to visit. Ajax found that out to their cost last month, while Barcelona were very relieved to escape with a narrow victory just before.
7.28pmMilan meanwhile have failed to win any of their last seven games in all competitions, leading to angry protests at the San Siro from the club's Ultras last weekend. Allegri is on borrowed time if you believe persistent speculation in the Italian media, which surely makes qualification for the knockout stages absolutely essential.
7.31pmBENCH WATCH! With tonight's match so pivotal in the Group H shake-up expect options in reserve to become very important late on, so it's well worth examining who Lennon and Allegri could call upon. Milan have Matri and Robinho bidding to prove their worth if they need a late goal, while Celtic have Anthony Stokes and Joe Ledley feeling hard done by to miss out on the starting lineup. A full look at both benches will follow shortly.
7.38pmHere are a few words from Lennon just prior to kickoff. "I've picked a very positive lineup because we need to try and win the game. I'm hoping Boerrigter can make a difference for us on the left-hand side and light the place up."
7.41pmPREDICTION! With Milan struggling in Serie A I expect the visitors to sit deep and wait for Celtic to force the issue tonight, a tactic that could come back to haunt them. Celtic deserve their strong European reputation at home and I think that could carry them to victory tonight, with Commons to get on the scoresheet in 2-1 win.
7.44pmA thunderous rendition of You'll Never Walk Alone sets the scene for a wonderful occasion, as Montolivo and Samaras lead out the two teams. Can Celtic keep their Champions League campaign alive? We are about to find out!
0 minHere we go then, with Milan getting this one underway shooting from left to right.
3 minMilan quickly zip the ball through to Balotelli, but he is well offside. The visitors soon find Balotelli again, but he is blocked off by Ambrose as he stride forward into the final third. A switch to Samaras ends up with Celtic winning an early corner kick, which sees Boerrigter miscue at the far post after losing his marker.
6 minAmbrose shows the other side of his game after missing a long goal kick with a rash leap, allowing Balotelli to take aim and smash wide from distance. A fast and furious start continues with Commons breaking inside the opposition half, before being closed down by two Milan midfielders.
9 minBoerrigter's early cross takes a nick and nearly catches Abbiati out of position, with Milan scrambling clear to stop Samaras seizing on a chance at the near post. Commons then makes a dash and forces Bonera to deal with a difficult long ball, but he heads back to Allegri comfortably.
11 minStrong hold-up play from Balotelli sees Milan win a corner kick, with Birsa's cross cleared behind by Ambrose. From the corner Birsa's delivery is a poor one, clearing a packed penalty area and allowing the hosts to regroup.
13 minKayal wastes a half-chance after beating the offside trap to control a wayward shot from distance, but just as he's about to pull the trigger Zapata puts the midfielder enough pressure to send a tame strike wide of the target. Celtic are starting to build momentum now.
17 minAdvantage Milan as Kaka tucks up a glancing header from close range, left completely unmarked to meet Birsa's cross and put the visitors ahead. Lennon will be furious will the defending from that set piece.
20 minMilan briefly lose Abate to injury after a clash with Kayal, but he returns Zapata hooks away a hopeful cross into the penalty area. Celtic will need more quality in the final third to get back in this match.
23 minForster comes off his line to claim a long pass from Birsa intended for Balotelli, although Milan's star striker is soon in possession as Milan recycle play. Worrying times for Celtic, who are choosing to play a risky game with a high, pressing line.
26 minEnd-to-end action as Boerrigter just misses the target for Celtic with a header from close range, after rising to meet Forrest's cross to the far post. Milan clear and quickly find Balotelli in the final third, with his rasping drive from 12 yards flying over the crossbar.
29 minBonera's vital interception stops Samaras being played free at the far post, but once again Celtic are defending just moments after threatening as Ambrose hooks away under pressure.
32 minSHOT! Celtic back off Kaka from 25 yards and nearly pay a very heavy price, with the Brazilian swerving a dangerous drive inches wide from distance. Before play can resume Kayal is gone on the ground and in some discomfort, looking very unlikely to continue.
34 minSUBSTITUTION: Kayal is forced off as he limps to the touchline, replaced by Joe Ledley. The Welsh international was a surprising omission and will desperate to prove a point and help the Hoops draw level.
37 minForrest is penalised for an over-zealous challenge in midfield, catching on Emanuelson, just moments after deflecting Boerrigter's drive well wide of the far post. Lennon is annoyed with the decision and makes his feeling clear with a show of dissent on the touchline.
40 minMulgrew forces Abbiati to make a save after a deflection off Birsa fell kindly into his path. From 12 yards the effort is a tame one though, allowing the Milan stopper to claim comfortably.
43 minSamaras commits a foul after pushing Zapata but is furious with the decision, earning a warning from the referee for his protests. Celtic are doing their best to press all over the pitch, but Milan look fairly comfortable at the moment.
45 min+1Samaras sends a glancing header through to Abbiati from a Boerrigter cross, anticipating the pass quickly but failing to generate enough power on the effort.
45 min+2Commons completely misses the penalty area with a lopping free kick and that should just about bring to an end to a frustrating first half for the hosts.
8.37pmMilan are in control of the match at the break, profiting from slack marking to break the deadlock courtesy of a Kaka header. Celtic have been more than a match for the hosts without forcing any genuine clear-cut opportunities, something that will have to change in the second half to keep their Champions League ambitions alive.
8.40pmThe visitors have looked solid and compact, while carrying a threat through Balotelli on the break. Allegri has gone for a more organised look tonight and so far reaped the rewards, with De Jong, Poli and Montolivo proving an effective shield in midfield.
8.43pmGROUP H UPDATE! We have news of a huge scoreline in tonight's other game in Group H, with Ajax holding a shock 2-0 lead over leaders Barcelona at the break. Unless the Catalan giants can respond after the break Celtic have to score against Milan, or they will be almost certainly be out of the competition.
8.46pmBENCH WATCH: Celtic have already been forced to bring on Ledley but may well need a super-sub to turn the tide tonight. Anthony Stokes and Amido Balde over very different attacking alternatives for the Hoops, while Milan have Matri and Robinho capable of taking over from Balotelli in attack.
45 minWe are back underway at Celtic Park, with the pressure firmly on Lennon's side to respond after being caught cold by Kaka early on. They quickly look long for Samaras, but Zapata is up strongly to clear.
48 minSHOT! Lustig draws a foul from Emanuelson and wins a free kick just outside the penalty area, giving Celtic a promising opportunity to test Abbiati. A blocked set piece eventually loops through to Van Dijk at the far post, but completely unmarked he can only fire straight at Abbiati. What an opportunity missed for the Bhoys!
52 minMilan edge further ahead and once again it's a tale of woe for Celtic defending from set pieces. Birsa's looping corner at the far post fell to Nocerino, who replaced Abata moments ago. His pass across goal found Zapata, who tapped home from close range.
55 minCeltic briefly think they have reduced the deficit almost instantly Van Dijk, but his celebrations are shortlived after sending Ledley's cross into the far post with a lopping header. An uphill battle now for the Hoops.
57 minGROUP H UPDATE! We have news of a goal in the game in Amsterdam, with Xavi bringing Barca back to 2-1. Ajax are also down to 10 men. Back in Glasgow Commons has been cautioned.
63 minIt's all over surely at Celtic Park as Balotelli puts the visitors three to the good. The striker controlled a brilliant long ball from Montolivo before holding off his marker with brute strength. Just as the ball appeared to get stuck under his feet Balotelli fired, sending a low shot beyond Forster and inside the far post.
66 minYELLOW CARDS! Van Dijk lets frustration get the better of him and slides through the back of Balotelli, picking up a caution for his efforts. Moments later the forward is hacked down again by Izaguirre, who also gets booked and will miss Celtic's final group game in Barcelona.
69 minSUBSTITUTION: Celtic swap Commons for Stokes as they desperately search for a late response, with Milan clearing a corner kick soon after the Irishman is introduced.
72 minSUBSTITUTION: Milan make a conservative change as they look to shut up shop, with Kevin Constant coming on for Birsa for a further defensive option.
75 minBoerrigter wins a corner kick with a bright run down the left flank, but as the Hoops do well to keep a cleared cross alive Ledley is crowded out in the penalty area. Celtic have another corner to show for their efforts though, which they take short.
78 minSamaras looks perplexed after being penalised for pushing Emanuelson over near the corner flag, summing up a frustrating night for the Greek striker.
81 minBoerrigter is the latest to he harshly penalised for a shove on Nocerino, as Celtic concede another free kick. Before Milan can resume both teams opt to make a change, with Robinho and Rogic on for Kaka and Forrest respectively.
84 minRobinho's lofted cross towards the far post evades Nocerino, allowing Celtic to stride out at the other end before Stokes casually loses possession. The hosts do earn a chance to whip another ball in the box though, as Bonera blocks Samaras to concede the foul.
87 minRobinho is down desperately calling to the bench after taking a heavy fall under pressure from Ambrose, who definitely shoved the Brazilian to the ground. Robinho looks to be struggling with an injured shoulder as he trudges to the touchline.
90 minAbbiati palms a looping free kick from Mulgrew to turn away from danger at the expense of a corner kick. In fairness to Celtic they have kept plugging away despite the writing being on the wall early in the second half.
9.39pmHeartbreak and frustration for Celtic then, with their Champions League dream ending this evening with a comprehensive home defeat. With Ajax beating Barcelona that means Neil Lennon's side can no longer qualify for the last 16.
9.42pmThat just about does it for Sports Mole's live text commentary from Celtic Park, but analysis of Milan's morale-boosting win will follow shortly. Goodnight!