A costly moment of slack play in the 22nd minute allowed French star Kingsley Coman to open the scoring and blight an encouraging first-half performance by Brendan Rodgers’ side.
However, Celtic roared back after the break and Callum McGregor deservedly levelled in the 74th only for midfielder Javi Martinez to restore the German giants’ lead three minutes later meaning that with two Group B fixtures remaining, the Scottish champions have only three points from 12 and cannot get into the knockout stages.
Bayern beat the Scottish champions 3-0 with ease in the Allianz Arena earlier in the month but the Hoops can take encouragement from this performance as they still remain on course for a place in the Europa League.
Bayern’s selection problems had dominated the build-up to the game after Poland striker Robert Lewandowski had joined Juan Bernat, Manuel Neuer, Thomas Muller and Franck Ribery on the sidelines with a thigh injury.
The solution coach Jupp Heynckes spoke of among six changes was to play James Rodriguez as a withdrawn main striker.
Celtic’s injury problems were again in central defence and with Jozo Simunovic and Erik Sviatchenko still out, midfielder Nir Bitton partnered Dedryck Boyata.
French striker Moussa Dembele started for the first time in the Champions League this season, supported by McGregor, as Rodgers returned seven of his big-hitters to the starting line-up.
Hoops midfielder Stuart Armstrong missed a sitter in the fifth minute, mis-kicking six yards out after James Forrest had set him up with a perfect cross.
The Bundesliga leaders were struggling to get into any sort of rhythm and in the 15th minute Dembele outpaced Rafinha in the left channel but dragged his left-footed drive wide of the far post.
Celtic’s good early work was undone, however, when Boyata hesitated and let a long punt from Bayern keeper Sven Ulreich go over his head.
Coman was on to the ball in a flash and rounded keeper Craig Gordon before slipping the ball past the backtracking Boyata, Bitton and Mikael Lustig.
On the half-hour mark Bayern midfielder David Alaba whipped the ball away from Dembele as he looked to tap in Armstrong’s pass across goal.
The former Dundee United midfielder came close three minutes after the restart with a drive which was brilliantly tipped over by Ulreich for a corner.
Amid great encouragement from the home fans, Rodgers’ side kept up their efforts and their goal arrived when Forrest threaded a ball through to McGregor, who drilled his shot low past Ulreich.
Celtic Park was still rocking when Coman took an Alaba pass and crossed for Martinez to rise above Bitton and head past Gordon.
It was a stunning blow to the home side but again they came back, fighting and scrapping for another leveller, only to run out of time before they were given a heartfelt round of applause from their supporters at the end.
Gordon, Lustig, Bitton, Boyata, Tierney, Brown, Armstrong (Griffiths 79), Forrest, McGregor, Sinclair (Rogic 64), Dembele.
Ulreich, Rafinha, Sule, Boateng, Alaba, Tolisso (Kimmich 83), Javi Martinez, Robben (Thiago 90), Vidal (Rudy 59), Coman, Rodriguez.
Danny Makkelie (Netherlands).