Goals by Mads Pedersen and Andre Hahn put Augsburg 2-0 up and even though Robert Lewandowski claimed his 14th goal in 12 games, it was not enough to rescue Bayern.
The Bundesliga leaders suffered only their second league defeat this season and their third in all competitions while Augsburg escaped the bottom three.
Second-placed Borussia Dortmund can now trim Bayern's four-point lead when they host Stuttgart on Saturday.
Thick fog shrouded the Augsburg Arena, but the hosts were crystal clear with their game plan.
Brazilian left-back Iago caused havoc on the flank as Augsburg, who were third from bottom before kick-off, raced into a deserved 2-0 lead.
Iago floated in a cross to Swiss striker Andi Zeqiri, who laid the ball off to Pedersen to drill his first Bundesliga goal past Manuel Neuer after 23 minutes.