Cologne, who defeated Bremen by seven goals on Saturday, continued their pressure on Bayern and won a corner kick after just three minutes.
Julian Chabot, a defender for Cologne, kept the ball in the air from the corner and found Ellyes Skhiri unmarked at the far post, where the Tunisian international scored from close range.
After 29 minutes, the visitors almost doubled their lead, but Bayern goalkeeper Yann Sommer made a fantastic double save to keep out Skhiri and Denis Huseinbasic.
At halftime, Munich manager Julian Nagelsmann made two substitutions, swapping out Germany internationals Leon Goretzka and Serge Gnabry for Kingsley Coman and Ryan Gravenberch.
Coman came close to making the change in the 53rd minute, so the change almost had an immediate effect. Marvin Schwaebe, the goalkeeper for Cologne, made a superb low stop to keep him out.
Cologne appeared to be on track to win for the first time since 2009 at Bayern, but a goal by midfielder Kimmich in the final minute of regulation time gave the hosts a point.
Aside from that, Wolfsburg continued their stunning recent form with a 5-0 victory over helpless Hertha Berlin, just days after scoring six goals to defeat Freiburg.
After just four minutes, Swedish midfielder Mattias Svanberg converted a free kick with a volley, and captain Max Arnold converted a flagrant handball by Hertha’s Dodi Lukebakio for his team’s second goal after 30 minutes.
Three minutes later, Jonas Wind scored his third goal in four days by curling in a third. The Wolves have now won six consecutive games thanks to goals from Ridle Baku and Omar Marmoush in the second half.
Hertha is last in the Bundesliga and has lost their last six games. On Saturday, Hertha plays Berlin against Union, who are fourth and have a game in hand over leaders Bayern.
In the fourth minute of extra time, Andre Kramaric of Hoffenheim scored a stunning long-range goal to help his team draw 2-2 with Stuttgart at home.
Kramaric gave Hoffenheim the early lead, but Serhou Guirassy and Wataru Endo’s goals seemed to put Stuttgart on the way to a crucial three points.