Four days after lifting the Club World Cup for a sixth title in less than nine months, Bayern crashed back to Bundesliga reality with Arminia racing to a 2-0 lead in the first half.
Bielefeld doubled their advantage in the 37th with unmarked Amos Pieper's powerful header from a corner and Bayern struggling in heavy snowfall.
Bayern, who had won eight of their nine previous league games, shot out of the blocks after the restart and Leroy Sane hit the post before Robert Lewandowski, with his back to the goal, turned quickly and volleyed in his 25th goal of the campaign in the 48th minute.
With former Bayern president Uli Hoeness still celebrating in the stands, Bielefeld restored their two-goal lead the very next minute. Winger Christian Gebauer tapped in a cutback with Bayern's defence caught out in the quick break.
Following the match, Bayern boss Hansi Flick said: "We did not defend well in the first half. The pitch was not ideal for our game," Flick said. "But in the second half the team showed its mental strength. That was outstanding.
"We have to say that today we can be satisfied with the 3-3."
Bayern are on 49 points, with RB Leipzig in second place on 44. Bielefeld are 16th on 18 points.