Bayern should have further punished Celtic as Craig Gordon pulled off a series of second-half saves, but the visitors were unlucky not to score themselves when Scott Sinclair saw an 88th-minute goal disallowed.
Relive how the action unfolded with Sports Mole's minute-by-minute commentary.
6.45pmGood evening all. Tottenham showed the blueprint last night for going to a European giant and coming away with a result. Can Celtic produce a performance at Bayern Munich which would be even more famous tonight? Let's get to the team news...
6.51pmCeltic are handed a huge boost with the return of inspirational captain Scott Brown, who has missed their last two matches with the injury which forced him off in the second half of the 3-0 win over Anderlecht. Stuart Armstrong, who hasn't played since the Old Firm derby in September, is also fit to take his place in the visitors' five-man midfield
6.53pmOne Celtic player who has not shaken off an injury is Bosnian defender Jozo Simunovic, who is struggling with a hamstring injury. Mikael Lustig is shifted inside to partner Dedryck Boyata in central defence, with Cristian Gamboa given only a second start of the season at right-back.
6.55pmBrendan Rodgers recalls Craig Gordon in goal after resting him for Saturday's 1-0 victory over Dundee United. Dorus de Vries, Nir Biton, Callum McGregor drop out and Eboue Kouassi misses out on the squad completely to accommodate the return of some key players.
6.57pmThe three men who scored in Brussels last month make up Celtic's front three - Leigh Griffiths, Patrick Roberts and Scott Sinclair - while Moussa Dembele, who is slowly returning to match sharpness, presents Rodgers with a great option from the bench if Celtic are still in the game late on.
6.59pmBayern Munich make just the one change to the team which hammered Freiburg 5-0 at the weekend, with the injured Javi Martinez replaced by Sebastian Rudy in a change which sees Joshua Kimmich drop back into defence.
7.01pmSven Ulreich continues to deputise for Manuel Neuer in between the posts, with the world's best goalkeeper on the treatment table along with Franck Ribery and Juan Bernat. Arturo Vidal has recovered from a knock sustained on international duty to take his place on the bench.
7.03pmArjen Robben makes his 100th Champions League appearance as part of a devastating front four which also includes Robert Lewandowski, who is in explosive form having scored 18 goals in 16 games for club and country this season.
7.05pmBayern's bench is not as strong as it might be had they not been missing several key players through injury, but James Rodriguez is among their substitutes, fresh off helping Colombia qualify for the World Cup.
7.07pmBayern sacked Carlo Ancelotti following their 3-0 defeat to PSG in their last group game, and have since reappointed Jupp Heynckes as caretaker manager until the end of the season. The 72-year-old's fourth spell in charge began in emphatic fashion with Saturday's dispatching of Freiburg.
7.09pmThe last Champions League game that Heynckes managed was the 2012-13 final at Wembley, when Bayern beat German rivals Borussia Dortmund 2-1 in extra time to complete a famous treble, before their coach stepped aside for Pep Guardiola.
The way that Manchester City are playing, it would be brilliant to see Guardiola return to Munich later in this year's competition.
7.11pmHowever, even during that victorious season, Bayern showed that they are vulnerable to a European giant-killing as they slipped up to BATE Borisov in the group stages. They also lost 2-0 to Arsenal in the first knockout round before going through on away goals.
7.13pmThat being said, the Bavarians have been absolutely imperious at home in Europe in recent years. They have won 22 out of 23 group games on their own patch, including 10 in a row since a 3-2 loss to Manchester City in 2013. The Allianz Arena is truly a European fortress which could humble Celtic.
7.15pmBayern have won their last four games against Scottish visitors to Munich, most recently a 5-1 home win over Aberdeen in the 2007-08 UEFA Cup for a 7-3 aggregate victory. They are unbeaten at home against clubs from Scotland with five wins and three draws.
7.17pmDespite this record and Bayern's tag as overwhelming favourites, Heynckes has insisted that he is taking nothing for granted.
He told reporters: "You see the transfer window and how crazy it has become...clubs like Celtic and others like Anderlecht can't keep up. But I would never make the mistake of under-estimating Celtic.
"Celtic are a physical, aggressive team. They have a lot of experience and they have won the Scottish Premiership six times in a row. They also have a great coach in Brendan Rodgers and have a great history in Europe but we want to qualify for the next round."
7.19pmHeynckes paints a pretty accurate picture with his pre-match comments. When this group was drawn it was assumed that PSG and Bayern would run away with it and that Celtic and Anderlecht would end up in a fight for third. Celtic and Bayern are currently level on three points, but only a huge upset from Brendan Rodgers' boys tonight would challenge that opinion.
7.21pmRESULT: It could be a night for European upsets, though! In tonight's early kickoff, Azerbaijani minnows Qarabag held Atletico Madrid to a 1-1 draw despite having a player sent off in the second half.
7.23pmCeltic should enter this game with a nothing-to-lose attitude. They will be full of confidence having extended their unbeaten domestic run to 59 matches (after a scare against Hibs) and picked up a morale-boosting victory at Anderlecht on matchday two.
7.25pmTheir victory in Brussels, which came courtesy of goals from Leigh Griffiths, Patrick Roberts and Scott Sinclair, was Celtic's first in 17 group stage games in either the Champions League or Europa League. The previous 17 had yielded seven draws and nine defeats.
7.27pmCeltic have never won away to a Bundesliga club - three draws and seven defeats - and are on a seven-game winless run, home and away, against German teams since beating Stuttgart 3-1 on their way to the 2002-03 UEFA Cup final, where they lost to Porto.
7.29pmHowever, the Bhoys have just shaken one European hoodoo, ending a run of five years without an away victory on the continent. Prior to getting the better of the Belgian champions, a 3-2 win at Spartak Moscow in 2012 was the last time Celtic had won away in Europe.
7.31pmRodgers has overseen significant improvement at Celtic, but when they have been tested against the very best teams in Europe, the Scots have been completely outclassed and embarrassed, losing 7-0 to Barcelona and 5-0 to PSG.
7.33pmThe former Liverpool manager has encouraged his team to take a more attacking approach tonight to finally do themselves justice against one of the Champions League heavyweight.
"We have a duty to play our game," Rodgers told The Guardian. "That is what we have already spoken of. Our disappointment against Barcelona and PSG was that we didn't quite impose our way of working. So our message is we have a duty to play how we play and that is to attack and be aggressive."
7.37pmHEAD TO HEAD: These two teams have only met twice previously - in the 2003-04 Champions League group stages. The Bhoys held Bayern to a 0-0 draw at Parkhead, but could not quite hang with their hosts in Munich as they slipped to a 2-1 defeat.
7.39pmIt must be said that a repeat of that result tonight might be considered a minor success for Celtic. That was a Bayern team that surrendered the Bundesliga to Werder Bremen in the same season, not one which has won five straight titles. The gulf between the two sides is much greater now than it was then.
7.41pmThis travelling contingent of Celtic fans has made its way into the stadium and is rocking ahead of one of the truly great European away days.
7.43pmPREDICTION: I'm going for 3-0 Bayern. Celtic were brilliant in Anderlecht but Bayern are an entirely different animal and the visitors will not be able to cope with Munich's forward four. I hope Celtic attack their illustrious hosts rather than sit back and accept their beating.
1 minKICKOFF: The famous music fades out and Bayern get the action underway...
3 minBayern have started rapidly, winning an early corner which Celtic clear. The promising sign is for the visitors is that, when they broke from the Munich set piece, five bodies were charging forward and they got a corner of their own.
4 minBrown stays down after being caught on the foot by fellow captain Thomas Muller. Celtic's all-time leading appearance maker in European competition gets back to his feet and will try to run this off.
6 minDISALLOWED! Thiago has the ball in the net but it won't count, because Lewandowski ran the ball out of play before crossing for the midfielder to emphatically finish from 12 yards. Replays show it was a close call whether the whole ball had crossed the byline.
8 minBayern want a penalty as Lewandowski is tugged to ground by Lustig. The Celtic defender definitely had a hold of his shirt but gets away with it. That is twice already the Swede has been caught out by a Lewandowski run into the right channel.
10 minThe hosts have Celtic under the cosh but a courageous run from Roberts gives them some brief breathing space. The youngster takes on a couple of players before David Alaba pinches the ball from him. Celtic are going to need more of that from Roberts and Sinclair to get them up the pitch.
12 minCeltic break two-on-two as Armstrong feeds Griffiths, who tries to cut inside to get a shot away on his left foot only for Rudy to get a tackle in, and the loose ball runs away from Sinclair.
13 minGood defensive work from Roberts this time as he tracks back to thwart Coman, who had cut inside Gamboa in the Celtic area. The ball eventually breaks to Kimmich but a wild shot from the right edge of the box flies wide.
14 minMISS! Coman darts past Gamboa on the Bayern left and fizzes in a cross which Brown clears, but only as far as Alaba who blazes horribly over the crossbar from 12 yards out.
16 minAnother good run from Roberts, who gets down the outside of Hummels and into the box, but his shot is blocked by Boateng. Again, there were three Celtic bodies up in support which shows the visitors are going to try to make the most of their counter-attacking opportunities.
17 minLewandowski, played onside by Gamboa directs a free header from eight yards out towards the bottom corner. Gordon makes a fantastic diving save, but Muller eats up the rebound for just his second goal of the season.
19 minCeltic just needed to weather the Bayern storm for another 5-10 minutes to try to get a foothold in the game. There is still a long way to go in this first half now and Celtic need to still be competitive by the break.
21 minThe Celtic defence is asleep as Kimmich fizzes a low corner to Lewandowski on the edge of the box, but the prolific Polish striker cannot keep his first-time shot down.
23 minYoung Kieran Tierney is furious as he advances down the left into a pack of Bayern shirts, signalling that there was no movement ahead of him. Celtic's front three have to remain brave because this could be a rout if the visitors can't keep the ball up the field for periods.
24 minSHOT! Coman drifts in from the left and unleashes a powerful low drive from 25 yards which Gordon does well to hold.
26 minAlaba swings in an early cross which Lewandowski glances well wide. The attempt was never troubling, but that he jumped above Lustig so easily to win the header might be a concern to Brendan Rodgers.
29 minBayern double their advantage as Coman skips past a helpless Gamboa and stands a cross up to the far post, where Kimmich meets it with a looping header back across goal and into the top corner. That is a finish that any striker would be proud of.
31 minThat is actually Kimmich's third goal in his last four games - not bad for a defender! Bayern really do have threats all over the pitch and this could get messy for Celtic is they are not careful.
33 minRoberts and Sinclair have tried their best to double up with their full-backs, but when Bayern get the ball in wide areas in wide situations, Tierney and Gamboa, in particular, are having a very tough time.
35 minBayern are popping the ball around with real confidence, instilled in them by new manager Heynckes. In this form they will be much more of a match for PSG and the rest of Europe's big boys when the time comes.
37 minUPDATE: Speaking of PSG, the free-scoring French outfit also lead tonight, against Anderlecht in Brussels thanks to Kylian Mbappe's early strike. So it is business as usual in Group B.
39 minYELLOW! Patrick Roberts is the first name in the referee's book tonight. The Manchester City loanee is annoyed after giving the ball away with a poor first touch and jumps in on Thiago. It's a very harsh booking though. He did get the ball and it might have been waved away in the SPL.
41 minCeltic need to get to half time with no more damage done. They look out of their depth here but if they can get to 60-65 minutes at only a 2-0 deficit and then pinch a goal on the break, you never know. Or, that could just anger the Bayern beast into going through the gears.
43 minComan again gets outside Gamboa and squares into the six-yard box but Boyata swings his boot through the ball to clear. If Coman will let Gamboa out of his pocket, Rodgers might want to spare the Costa Rican more punishment in the second half and stick Lustig in at right-back.
8.37pmBayern have enjoyed total domination here and Celtic are lucky to go into the break only 2-0 down. They have been all over the place defensively and not posed an attacking threat whatsoever. The gulf in class is just too big, I'm afraid.
8.39pmA quick recap: after surviving an early disallowed goal and penalty appeal, Celtic went down to Muller's 17th-minute opener and conceded again before the half-hour mark through Kimmich's superb header.
8.41pmBayern look like a completely different side to the one which was dismantled by PSG last month. They have quickly rediscovered their confidence under Heynckes and are determined to show they are still a force in this competition.
8.43pmTheir most productive outlet has been Coman on the left flank. When he has been able to isolate Gamboa, he has had his way with the Costa Rican and Rodgers might want to consider a change there.
8.45pmUPDATE: PSG lead Anderlecht 2-0 in Group B's other game, and Chelsea have also scored twice to lead Roma 2-1 at the break. Manchester United are putting in a classic Jose Mourinho away performance in Benfica (0-0). For all of tonight's latest scores, click here.
8.47pmCeltic face the choice now of embracing this mountainous challenge of trying to get back into the contest, or simply playing damage limitation. If they concede one more, I expect the Scots to go into their shells.
46 minKICKOFF: The action resumes at the Allianz Arena, with both sides unchanged...
47 minRoberts creates a yard of space to bend a left-footed shot around David Alaba, but his shot flies comfortably wide. Nevertheless, that represents Celtic's best opening of the encounter so far.
49 minLustig does really well to slide in and intercept a through ball which would have put Lewandowski through on goal. At the other end, Roberts is seeing more of the ball and is trying to make something happen for Celtic.
51 minHummels runs off Boyata to meet Robben's inswinging corner with an inch-perfect header into the side-netting. That all stemmed from a sloppy pass from Gordon which led to the concession of the corner. Game over, and Celtic are in danger of self destructing here.
53 minSHOT! Celtic test Ulreich for the first time this evening as Tierney weaves his way through several bodies by the left byline and pokes a shot which the Bayern deputy holds comfortably.
55 minLewandowski looks to add his name to the scoresheet with a 25-yard free kick, but the Celtic wall gets in the way.
57 minIt remains all Bayern as they push Celtic deeper and deeper into their own territory. The two Munich full-backs are playing like wingers and it is creating an overload which the visitors just can't deal with.
59 minRobben goes down the outside and tries to tee up Lewandowski in the middle, but Gordon is off his line to prevent Bayern from getting an easy fourth.
60 minLovely pass from Roberts to pick out the out-to-in run of Sinclair, but Kimmich followed him all the way and when the ball breaks to Ntcham the shot from the Frenchman ends up in the second tier.
61 minOFF THE LINE: Gamboa clears the ball from under his own crossbar! Robben was the unlikely player to meet Kimmich's corner and his header had Gordon beat before the full-back intervened.
62 minDISALLOWED! Bayern are denied by assistant referee for a second time after Lewandowski cushions Alaba's pinpoint cross past Gordon from close range. The assistant got that one right, in spite of the Pole's protests.
64 minMuller and Robben link up beautifully on the halfway line and the Dutchman tries to find his teammate with a reverse pass which a retreating Brown intercepts. No matter how bad it gets, the Celtic captain will not give up, that is a guarantee.
65 minSUBSTITUTION: Double change from Rodgers as Tom Rogic and Moussa Dembele come on for Griffiths and Armstrong. It was probably the wrong decision to start Armstrong over Rogic as it has given Celtic no greater defensive protection and the Bhoys have missed Rogic's hold-up play.
67 minSUBSTITUTION: With the job done, Heynckes brings off Thiago for Arturo Vidal in midfield. No respite for Celtic with a player of that international calibre coming off the bench with still a quarter to play.
69 minRogic spins and feeds Tierney down the left but the cross into the near post is cut out by Boateng.
70 minWOODWORK! Out of nowhere, Tierney unleashes a 30-yard drive which might have been heading for the top corner before Ulreich tipped it onto the post. Great effort from the full-back, and Celtic don't even get a corner for their troubles.
73 minSAVES! Vidal passes to Robben inside the box, but Gordon spots him opening up his body for a left-footed shot and makes a good save to his right. Moments later, Lewandowski is up at the back post but cannot generate enough power on his header to trouble the goalkeeper.
75 minCHANCE! Celtic go close to pulling one back as Rogic sets up Sinclair in the left channel for a drive which Ulreich blocks. The Bayern stopper is up quickly to save the follow-up effort from Rogic, too. A goal would be a nice consolation for the travelling Celtic fans.
77 minSUBSTITUTION: Three-nil down and on comes James Rodriguez for Bayern, although Celtic will be happy to see the back of Coman who has absolutely terrorised Gamboa all evening.
80 minSUBSTITUTION: Each manager makes their final change, with Rafinha coming on for Kimmich and James Forrest replacing Roberts, who has probably been Celtic's best player, although that is not saying a lot.
82 minSHOT! James Forrest charges down the right and shoots powerfully from an angle, with Ulreich parrying behind for only Celtic's second corner of the match.
84 minBayern have entered cruise mode here, knocking the ball around in midfield comfortably without showing too much intent to get a fourth. Lewandowski would love to get in on the action though.
86 minDISALLOWED! Sinclair curls the ball around Rafinha and beyond Ulreich to grab what he thinks is a Celtic consolation, but he is denied by the offside flag as Lewandowski was earlier this half.
88 minCHANCE! Vidal nearly adds a fourth but he fires his over-the-shoulder volley straight at Gordon, after being picked out by James's wonderful pass.
90 minIn the first of THREE minutes of stoppage time, Rogic wins the header from a corner and Brown tries to divert it into the net from close range. His effort hits the post and the offside flag spares the Celtic captain's blushes because he really should have scored.
90+2 minSAVE! We have seen some great saves this week and this is another one from Gordon, who stands tall to block Vidal's header from point-blank range and keep the difference at three.
9.39pmThe final whistle brings to an end a comprehensive defeat for Celtic, who were completely blown away by their illustrious opponents. It finished 3-0 but in truth it could have been a lot more.
9.42pmThat puts Celtic's European campaign into perspective and confirms the best they can hope for his third in Group B and a Europa League spot. Bayern and PSG are just in a different league and their fate will be decided by the rematch against Anderlecht at Parkhead.
9.45pmThat's it from me. If you're a Celtic fan who can withstand slightly more mental punishment, take a read of our match report from Munich. If you can't stand to read it, distract yourself with one of our other reports on tonight's Champions League action. Thanks for joining Sports Mole tonight!