Bayern joined Arsenal in sealing a top-eight finish in the 36-team table with a game to spare, avoiding a potential two-legged knockout play-off round.
The six-time European champions sit second in the standings behind Arsenal ahead of their last league-phase outing at PSV Eindhoven next week.
Kane headed in from a corner before winning and converting a penalty to put the hosts firmly in control.
The England captain had a chance for a hat-trick but missed a late spot-kick, after Bayern centre-back Kim Min-jae had been dismissed for a second yellow card.
“Coming back into the second half we wanted to show more intensity and I think we did that,” Kane said.
“We scored at the right times, were a bit unlucky with the red card but even after that when we went down to 10 men we controlled the game well and earned the three points.”
Bayern’s Champions League defeat at Arsenal remains their only loss in all competitions this season, in which they have also already opened up an 11-point lead in the Bundesliga.
Kane’s second goal was his 34th of the campaign, bringing Bayern’s total tally to 103 strikes from 29 games.
At Stamford Bridge, Moises Caicedo’s late goal gave Chelsea a vital 1-0 win against Cypriot minnows Pafos as the Blues moved closer to automatic qualification for the last 16.
In Liam Rosenior’s first Champions League game as Chelsea boss, Caicedo netted in the closing stages to end Pafos’ stubborn resistance. Chelsea’s fourth victory from seven Champions League group stage matches lifted them into eighth place.
Meanwhile, Fermin Lopez’s first-half brace and second-half goals by Dani Olmo and Robert Lewandowski earned Barcelona a 4-2 win at Slavia Prague.
The result keeps Barca in contention for a top-eight finish in the league phase, which guarantees direct passage to the last 16.
The Catalans face Copenhagen at home in their final league phase game next week.
“I’m happy that we came back and the team played really good and I also think at the end we deserved the points,” Barca coach Hansi Flick told reporters.
“We didn’t start well in this Champions League, but at the end we still have the chance to reach the (top) eight and this is what we want, because then we’ll have two games less,” he added.
At St James’ Park, Newcastle climbed into position to reach the last 16 by punishing PSV Eindhoven’s defensive blunders for a 3-0 win. (AFP)
