After 73 minutes, the England captain scored with a simple tap-in to put Bayern ahead 2-1 before setting up Jamal Musiala six minutes later.
Mauro Icardi of Argentina scored an outrageous Panenka penalty in the first half to offset Kingsley Coman’s eight-minute goal.
Bayern’s winning streak in group stage matches in the Champions League now stands at 16.
With three victories in three, they keep a commanding lead in Group A.
Bayern Munich took the lead in a back-and-forth contest when Musiala leapt over the ball with a deft fake to allow Leroy Sane’s pass to roll to Coman, who scored from the edge of the box.
Joshua Kimmich’s careless sliding foul on Icardi in the box resulted in a penalty for Galatasaray, who continued to press for an equalizer.
When the attacker dinked his penalty kick over Bayern goalkeeper Sven Ulreich at RAMS Park, the home crowd erupted in jubilant applause.
Just before halftime, Icardi should have given the Turkish team a second goal, but despite being unmarked, his effort went high and wide.
As his teammates struggled to identify him in the second half, Kane cut a furious figure. The hosts had a strong second-half start.
The former Tottenham striker attempted a daring backheel, which was diverted back into his path, leading to a straightforward finish and his second Champions League goal for the German champions. Nevertheless, it had little effect on his confidence.
When he pulled a pass back to the penalty spot for Musiala to rush on to and slip past Fernando Muslera, he was there in the middle of the action once more.
Deep into stoppage time, a despondent Galatasaray thought they had scored, but it was disallowed because Abdulkerim Bardakci was found to have used his arm to steer the ball toward the goal.