Bayern came into the clash on the back of a 2-0 win over Schalke, which extende their winning run to four matches this season.
The game got off to a lively start, with Bayern going close in the opening 15 minutes through Thomas Muller and Sandro Wagner, while Caiuby also squandered a chance for the visitors.
It was a first half full off missed chances for the defending champions, with Wagner being guilty of squandering a number of chances as the sides went into the break all square.
Arjen Robben got the game's opening goal shortly after the break with a clinical finish after rounding goalkeeper Andreas Luthe, following some good work by Serge Gnabry.
Ribery thought he had doubled Bayern's lead with six minutes remaining in the second half, but the Frenchman's goal was chalked off after Muller was flagged offside in the build-up.
Augsburg then silenced the Allianz Arena in the 86th minute, with substitute Felix Gotze slotting home from close range to seal a point.