Jack Byrne gave his side the lead after 26 minutes before Bayern goalkeeper David Hundertmark was dismissed ten minutes later for handling outside the area.
The irrepressible Jose Angel Pozo doubled the advantage two minutes from the break to emphasis the dominance the away side had enjoyed.
City’s best player on the day, Angelino, had already had a hand in both goals before he made it 3-0 on the hour mark himself, latching onto a brilliant pass by Brandon Barker and slotting home.
It was four courtesy of a Bersant Celina thunderbolt six minutes later, with the young Kosovan being set up by Olivier Ntcham.
Thierry Ambrose headed against the crossbar in the dying embers but it was Bayern who scored the game’s fifth and final goal in stoppage time, though in truth it was nothing more than a consolation.
Roma beat CSKA Moscow 3-1 in Group E’s other game, setting up a mouthwatering meeting between the Italians and Vieira’s men at Hyde on 30 September.
