Mergim Berisha slots in a cut with a composed finish in the 59th minute to seal the three points and put more pressure on the Bayern coach Julian Nagelsmann.
His team has now drawn three and lost one of their last four league games, failing to show the form that led them to dominate the league for a decade.
It was the first time in 87 games that Bayern had failed to score in a league game and the first time in 20 years that the team had played four league games without a win.
“After four games without a win, we are stunned and ashamed,” said Bayern striker Thomas Mueller. “I can’t talk about many positives about this game.
“If you really analyze it, we could be winners with so many opportunities we’ve created. But if you don’t convert them, you’re here with zero points.”
The hosts were not intimidated by Bayern at any stage and their seven shots on goal were almost double Bayern’s tally in the first half.
The Bavarians, fresh off their Champions League group win over Barcelona in the middle of the week, missed a clinical finish with both Sadio Mane and Leroy Sane having their fair share of chances.
Germany international Sane should have equalized in the 73rd when he only had Augsburg’s Rafal Gikiewicz to beat, but the keeper held out to block his shot.
Gikiewicz then made a sensational double save in stoppage time to protect Augsburg’s lead. First, he stopped a scoreless header from goalkeeper Manuel Neuer, who had come forward, and then blocked the rebound.
Bayern is on 12 points, three of leaders Borussia Dortmund. Augsburg climbed to 11th place at nine.