Bayern expands their lead at the top of the Bundesliga to five points over RB Leipzig and next up for them is a trip to Eintracht Frankfurt.
Meanwhile, Arminia is level on points with Hertha Berlin and could move out of the bottom three with a victory or draw against Wolfsburg next time out.
The referee calls time on a thrilling game at the Allianz Arena, which ends 3-3 despite the visitors twice holding a two-goal advantage.
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.
The visitors astonished Bayern with a goal in the ninth minute when Michel Vlap latched onto a pass, swung around and beat keeper Manuel Neuer with a well-placed shot.
Lewandowski cut their lead in half just three minutes after the restart, however, Arminia got a third less than a minute later.
Bielefeld increased their advantage in the 37th with unmarked Amos Pieper’s powerful header from a corner and Bayern battling 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.
With former Bayern president Uli Hoeness still celebrating in the stands, Bielefeld restored their two-goal lead the very next minute.
Winger Christian Gebauer wiped in a cutback with Bayern’s defence caught out in the quick break.
Bayern then bounced back and made it 3-3 after goals from Tolisso and Davies, but couldn’t get another to win the game.
Bayern is on 49 points, with RB Leipzig in second place on 44.
Bielefeld is 16th in the league standing.