The German giants needed a 90th minute strike from former City defender Jerome Boateng to land them all three points at the Allianz Arena, giving them a winning start to the new European campaign.
City were unlucky to be beaten in the first Group E outing but eventually Joe Hart's resistance was broken when Germany international Boateng, who spent time with the English champions in 2010/11, fired home from the left side of the box in the final minute of regular time.
Throughout the hard-fought contest, England keeper Hart had been at his best to deny the likes of Thomas Muller, Mario Gotze, David Alaba and Robert Lewandowski while City were desperately out of luck when David Silva thought he should have won a late penalty.
However, the killer blow was landed at the death when Boateng, who had moments earlier seen a stunning effort kept out by Hart, controlled the ball as it fell to him from a deflection and powered home with thanks from a little bobble off the back of teammate Mario Gotze.
It was a frustrating result for City, and especially Hart, who had been at the top of his game for almost an entire 90 minutes but came up short against a side which fielded six World Cup winners in their starting XI.
Pep Guardiola's Bayern will next meet Russian side CSKA Moscow on September 30 while Manuel Pellegrini's City will prepare for their home clash with Italian outfit AS Roma following their 5-1 triumph in Rome tonight.
UEFA Champions League - Group E
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