Bayern made three changes from the FIFA Club World Cup final, as Bouna Sarr, Corentin Tolisso and Eric Maxim Choupo-Moting were rotated into the line-up.
The game got off in snowy conditions and Arminia surprisingly went ahead in the ninth minute after Michel Vlap turned brilliantly on the edge of box from Manuel Prietl's pass and curled a strike past goalkeeper Manuel Neuer.
Choupo-Moting had a wonderful chance to level matters for the Bavarians in the 32nd minute after a through ball over the defence by David Alaba but his strike was saved by shot-stopper Stefan Moreno.
Amos Pieper doubled the lead for the visitors eight minutes before the break with a powerful header inside the box that found the top corner, leaving Bayern stunned after a corner by Vlap.
Robert Lewandowski pulled a goal back for the Bavarians three minutes from the break with a brilliant volley into the bottom corner after impressively controlling a cross over the top by Alaba.
Christian Gebauer restored the two-goal cushion for Arminia seconds later with a tap-in after Andreas Voglsammer beat Sarr on the left wing before making the vital cross.
Tolisso reduced the deficit again for Bayern just before the break with a powerful header around the penalty area after a teasing cross by Leroy Sane.
Alphonso Davies levelled matters for Flick's men in the last 20 minutes with a smashing strike from the rebound on the edge of the box that found the bottom corner.
The game was there for the taking for either side in the closing minutes, with Bayern forcing to snatch all the three points but it was not to be, as a draw left Bayern leading the table by five points while Arminia remained in the relegation play-off spot.
