Tom Bischof and Michael Olise also got on the scoresheet as Bayern struck back after trailing early on when the hosts struck through a superb Robin Meissner strike.
"It's never easy at an away ground with a very good atmosphere. They started the game with an amazing goal, and we had to stay calm," Kane said.
"I'm happy we got the win."
England captain Kane now has 148 goals in 150 games in all competitions for Bayern Munich, whom he joined from boyhood side Tottenham in 2023.
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Meissner had the hosts dreaming of a shock upset when he scored with a lovely dipping shot from outside the box after just five minutes.
Bayern, the 21-time German Cup winners, were last eliminated at the first round in 1994 and quickly levelled things up when Kane hammered home a rebound.
Bischof added another in similar fashion six minutes later to put Bayern in the box seat. Olise scored on 73 minutes and Kane converted a penalty with four minutes remaining to seal victory.
On Tuesday, Borussia Dortmund beat fifth-division amateur side HEBC 5-0 in Hamburg.