The England captain fired home his 18th goal of the season after 20 minutes to fire Bayern to the top of the Bundesliga table, at least until Bayer Leverkusen face Werder Bremen on Saturday.
In doing so, Kane became the first player in the competition's history to score 18 goals in the opening 12 games of a season, while in all competitions he has 22 goals in 17 appearances.
Even more remarkably, in just 12 matches in the German top flight, Kane has now scored more goals than any other Englishman has managed in an entire Bundesliga season before, overtaking the 17-goal mark previously jointly held by Kevin Keegan in 1978-79 and Jadon Sancho in 2019-20.
As well as lifting Bayern to the summit of the table, the result also sees Koln slip to the bottom in place of Union Berlin, having now won just one of their 12 games this season.
However, the contest was a closer one than many might have expected before the game, and Koln - the division's lowest scorers - even had a chance to take the lead when Jan Thielmann was denied by Manuel Neuer.
Such chances were few and far between for Steffen Baumgart's strugglers on the night, though, and Bayern were never likely to be as wasteful with the in-form Kane leading the line.
What proved to be the winning goal came after 20 minutes when the former Tottenham Hotspur man tucked home after Eric-Maxim Choupo-Moting's initial effort had been blocked on the line.
The goal means that Kane has now already surpassed Robert Lewandowski's goal tally for his first season at Bayern, while keeping him on track to break the Polish striker's all-time Bundesliga record of 41 goals in a single campaign.
Choupo-Moting should have doubled the advantage shortly afterwards, while Leroy Sane also squandered a presentable opening and Kingsley Coman rattled the crossbar as Bayern went in search of a second.
Just the one goal proved to be enough in the end, though, as Bayern made it six league wins in a row and maintained their unbeaten record so far this term.