Roberto Mancini's men struggled against the might of the German side, who dominated possession throughout to take a comfortable victory at the Allianz Arena.
Bayern's two came towards the end of the first half, with Mario Gomez finding the net from close range in the 38th minute before following up to make it a brace seven minutes later.
The Blues made their Champions League debut a fortnight ago with a 1-1 draw with Napoli at the Etihad. Mancini subsequently admitted that his side were "nervous" to be entering the European competition for the first time.
Bayern now sit atop group A with a maximum six points from their first two matches. Tonight's other game in the group ended with Napoli defeating visitors Villarreal 2-0.