Bayern fell behind for the first time in the season when Ragnar Ache headed courageous hosts Cologne 1-0 up off a corner kick from Isak Johannesson in the 31st minute on Wednesday (local time).
But Bayern needed a mere seven minutes to turn the second-round match on its head.
Luis Diaz tapped a 36th-minute equaliser after goalkeeper Ron-Robert Zieler had saved a first attempt from Josip Stanisic.
Bayern were lucky because Diaz was offside, which went unseen by the match officials, and video reviews are not used in the first two rounds of the competition.
There was no debate two minutes later when Kane got the ball with his back to goal, turned 180 degrees and fired past Zieler into the top far corner.
Diaz shot wide with only Zieler to beat after the restart, but Kane headed the third in the 64th minute off a corner kick from Joshua Kimmich for his 22nd club goal of the season, and Michael Olise wrapped up matters on the counter in the 72nd minute.
Bayern's 14th win moved them one ahead of Italy's AC Milan, who won 13 matches at the start of the 1992-93 campaign.
Bayern are record German Cup winners with 20 titles, but the last one came in the 2019-20 treble-winning season. They have not gone beyond the quarter-finals since then.
Bayer Leverkusen scored twice in the final minutes of extra time to win a thrilling game 4-2 over second-division Paderborn and book a spot in the third round.
In other games, defending German Cup champions Stuttgart beat Mainz for the second time in three days with a 2-0 win to reach the third round. Stuttgart also beat Mainz 2-1 in the Bundesliga on Sunday.
The lowest-ranked team left in the competition, fourth-tier Illertissen, were beaten 3-0 by second-tier Magdeburg.
Kaiserslautern won 1-0 at Greuther Furth, Freiburg beat second-tier Fortuna Duesseldorf 3-1, and Darmstadt crushed Schalke 4-0.
- with AP
