In the 20th minute, the England captain netted his 18th goal of the season, propelling Bayern to the top of the Bundesliga table, at least until Bayer Leverkusen faces Werder Bremen on Saturday.
Kane's feat of scoring 18 goals in the first 12 games of a season makes him the first player in the Bundesliga's history to achieve such a record. Across all competitions, he has amassed 22 goals in 17 appearances.
Impressively, within just 12 matches in the German top flight, Kane has surpassed the goal tally of any Englishman in an entire Bundesliga season, overtaking the previous joint record of 17 goals held by Kevin Keegan in 1978-79 and Jadon Sancho in 2019-20.
Beyond lifting Bayern to the league summit, the result also saw Koln drop to the bottom, replacing Union Berlin. Koln has now won only one of their 12 games this season.
While the match was expected to be one-sided, Koln, the division's lowest scorers, had a chance to take the lead when Jan Thielmann's attempt was thwarted by Manuel Neuer.
However, chances for Steffen Baumgart's struggling side were sparse, and with the in-form Kane leading the line, Bayern was unlikely to be wasteful. The decisive goal came in the 20th minute when Kane capitalized on Eric-Maxim Choupo-Moting's initial effort being blocked on the line.
Kane has now exceeded Robert Lewandowski's goal tally for his first season at Bayern, keeping him on course to break the Polish striker's all-time Bundesliga record of 41 goals in a single campaign.
Although Choupo-Moting and Leroy Sane missed opportunities to extend the lead, and Kingsley Coman struck the crossbar, the single goal proved sufficient as Bayern secured their sixth consecutive league win and maintained an unbeaten record this season.
