The defeat, which was fully-warranted after a dominant Bremen display, leaves Bayern seven points from leaders Bayer Leverkusen, albeit with a game in hand.
Mitchell Weiser's goal won the match after Bremen had already celebrated a potential winner, that was ruled out for a foul early in the build up.
The match also saw fans hit out at manager Thomas Tuchel, whose comments after their previous game against Hoffenheim went down like a lead balloon.
Supporters at the Allianz Arena had criticised a German league investment deal, to which Tuchel responded: "It's time for a home game again with enthusiasm."
Fans hit back at the comments with a banner in the north curve which read: "A curve also thrives on protest and resistance - enthusiasm doesn't come at the push of a button!"
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The result will further pile pressure on Tuchel, who took over from Julian Nagelsmann in March last year, and instantly led the team out of the Champions League and DFB Pokal.
Bayern still won the Bundesliga on the final day from Borussia Dortmund, but they face a far sterner task this time around against Xabi Alonso's Leverkusen.
Much of their struggles in 2022/23 were attributed to failing to replace Robert Lewandowski, but with £88million 26-goal Kane, their problems should have been solved.
However, the England captain, who is yet to win a trophy in his professional career, was left crestfallen after another poor result.
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His debut in the German Super Cup saw Bayern miss out on their first piece of silverware with a 3-0 defeat to Leipzig, and in November they were knocked out of the DFB Pokal again, this time by third tier Saarbrucken.
A record of 13 wins and 52 goals in 17 games would still be title-winning form for Bayern, but that's nothing on Leverkusen, who remain unbeaten in all competitions this season ahead of a huge clash against Bayern on February 10.