Augsburg went into Friday night's match unbeaten in their last 11 games and took the lead thanks to a bit of quick thinking after half an hour. With the Bayern defence static, Jeffrey Gouweleeuw took a quick free kick and Dimitrios Giannoulis controlled the ball with his instep before hammering it into the roof of the net for his first league goal.
However, the hosts were not ahead for long. Jamal Musiala made it 1-1 three minutes before halftime when he kept his feet in a crowded penalty box and fired home after good work from Leroy Sane on the left.
The game changed in the 59th minute when Cedric Zesiger was sent off for fouling Kane. A minute later, the Englishman scored with a diving header to put the visitors ahead. It was his 23rd league goal of the season.
An own goal from Chrislain Matsima made it 3-1 in added time.
"Augsburg away has always been a tough fixture for Bayern," coach Vincent Kompany said.
"We expected a difficult game. We stayed calm after conceding the opener and kept going. Overall, if you see how we played, it was a good away win, we take the three points."
The only bad news for Bayern was a second-half injury to Musiala. The forward limped from the field with what looked like a hamstring injury.
Bayern board member for sport Max Eberl told broadcasters DAZN "it doesn't look so brilliant" and ruled him out of the Champions League quarter-final first leg against Inter Milan on Tuesday.
"It's extremely bitter, regardless of whether it's Jamal or someone else. It's really blatant the amount of injured we have at the moment."
Bayer Leverkusen (59 points) can cut Bayern's lead back to six when they play away at third-bottom Heidenheim on Saturday.
Augsburg (39) remain eighth in the standings.
With DPA.
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