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Kane scored twice, in the 18th and 86th minutes, the latter from the penalty spot, while Tom Bischof (24) and Frenchman Michael Olise (73) added the other two goals for the Bavarian club, who responded strongly after falling behind early to a superb strike from Robin Meisner (5).
The England captain has now scored 148 goals in 150 appearances in all competitions for Bayern Munich, having joined the club from his boyhood side Tottenham in 2023.
Meisner gave the hosts hope of producing a major upset when he opened the scoring after just five minutes with a fine looping effort from outside the penalty area.
Bayern, who have won the German Cup 21 times, most recently last season against Stuttgart, quickly levelled when Kane pounced on a rebound and found the net.
The Bavarian side's last first-round exit from the competition came in 1994.
Bischof added a second goal in similar fashion six minutes later to put Bayern in a comfortable position. Olise then scored in the 73rd minute before Kane converted a penalty four minutes from time to seal the victory.
Borussia Dortmund had beaten fifth-tier amateur side HIBC 5-0 in Hamburg on Tuesday.