Boateng scored with a wonderful half-volley that took a slight deflection from Robert Lewandowski before flying inside the far post after Arjen Robben's corner.
It was looking like Manchester City goalkeeper Joe Hart's inspired performance would be enough to salvage a draw for the visitors, who went closest through David Silva's flying header in the 63rd minute.
It was the third meeting in four seasons between the sides, with Bayern avenging defeat at home to the Premier League side in their final group game last year.
In the other Group E game, Roma defeated CSKA Moscow 5-1 in Rome.
All content copyright ©2014 Daily Journal, a division of Home News Enterprises unless otherwise noted.