The Bavarians, who travel to defending champions Paris St Germain in the Champions League next week, were initially on the back foot, with Cologne carving out several good chances before Ragnar Ache headed in a corner for a deserved lead in the 31st.
Bayern, however, needed just five minutes to turn the game on its head, with Luis Diaz scoring on the rebound to level in the 36th, with the Colombian seemingly offside but with VAR unavailable in Cup games.
Top scorer Kane then completed their comeback in the 38th with a sensational second goal. The England captain got the ball with his back to the goal, turned and curled a left-footed shot into the top corner.
Diaz should have put the game to bed in the 53rd when he had only the Koln keeper to beat, but completely missed the target, firing high over the bar. Kane did it much better 11 minutes later, heading in from a corner to kill off the tie and bag his 22nd goal in 14 matches for Bayern this season.
Michael Olise tapped in from a Diaz assist in the 72nd to complete their win.
Thanks to the result, Bayern have become the first European side ever to start their season with 14 consecutive wins.