Kane headed home shortly before and after halftime, keeping up the good work started by first-half scorer Jamal Musiala.
Serge Gnabry got the fourth as Bayern seemed to be heading towards a huge home win.
But Finn Porath put Kiel on the scoreboard and two goals from Steven Skrzybski in stoppage time led to a thrilling finale which had appeared impossible half-an-hour earlier.
"No-one is happy that it ended 4-3. We can not switch off like this," Musiala told Sky TV.
Kane was playing his 50th Bundesliga match, and his tally of 55 goals in that period is a league record, five more than Erling Haaland managed for Borussia Dortmund in his first 50 games.
"It means a lot, means that you are doing something well. It is nice to have those numbers and I will keep pushing," Kane said.
Kane's productive day ended with news that he has also won the popular 2024 Goal of the Year award by broadcasters ARD for a Bundesliga goal he scored against Augsburg in November.
He controlled, with artistry, a long ball from Leon Goretzka with a kung-fu like move with his right foot, and then calmly headed home, with goalkeeper Nediljko Labrovic well beaten.
Elsewhere on Saturday, Dortmund claimed their first Bundesliga victory of the year at the fifth attempt, 2-1 at lowly Heidenheim, with goals from Serhou Guirassy and Maximilian Beier.
Interim coach Mike Tullberg celebrated wildly after he managed to steady the ship in his three-game spell after Nuri Sahin got the boot following defeats in the first four 2025 games in all competitions.
Tullberg has achieved two victories and a draw before Niko Kovac takes over on Sunday.
Dortmund are 10th but only three points off the top four.
Bayern now top the table nine points ahead of champions Bayer Leverkusen, who host struggling Hoffenheim on Sunday.
Union Berlin failed to take their chances in a 0-0 draw with Leipzig, illustrating why the club is looking for a forward to boost its attacking ranks.
Croatian forward Marin Ljubičić, who has been playing for Austrian team LASK, watched the match in Union's stadium ahead of an expected transfer.
The draw was enough to lift Leipzig to fourth place, the last spot for Champions League qualification, thanks to Borussia Mönchengladbach's 2-1 win in Stuttgart earlier. Leipzig moved a point ahead of Stuttgart.
Socceroo Jackson Irvine's St Pauli were held 1-1 by Augsburg, and bottom side Bochum lost 1-0 against Freiburg.
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