The Bavarians, who clinched this season's title in record time with seven games left, had not lost in the Bundesliga since a 2-1 defeat by Bayer Leverkusen in October 2012.
With Bayern coach Pep Guardiola fielding a second-string team and resting key players ahead of next week's Champions League quarter-final second leg against Manchester United, the champions had trouble finding their rhythm.
"We tried everything until the end," Guardiola told reporters. "But we have to accept that in football you cannot always expect to win. Now we have to prepare quickly for our big game against Manchester."
The Spaniard left Arjen Robben, Franck Ribery and captain Philipp Lahm back in Munich and fielded teenagers Mitchell Weiser, Ylli Salahi and Pierre Hojbjerg.
"We want to win every game and we wanted to do it today as well but it did not work," Bayern midfielder Toni Kroos said. "But we still do not want to give away the games."
Augsburg made a strong start and shocked the visitors with Moelders, starting instead of striker Raul Bobadilla, drilling in on the half hour after Mitchell Weiser lost possession in the Bayern half.
Augsburg briefly picked up where they left off early in the second half with Alexander Esswein's effort palmed away by keeper Manuel Neuer.
Guardiola sent on Thomas Mueller, Mario Goetze and David Alaba and the visitors instantly wrestled control away from the hosts.
Austria international Alaba hit the post as Bayern put Augsburg on the back foot and Goetze and Mueller also went close for the Bavarians.
Augsburg successfully soaked up the Bayern pressure and even hit the post in the final minute before claiming a memorable victory, their first ever in the league against a team leading the Bundesliga.
Schalke 04 joined second-placed Borussia Dortmund, in action against VfL Wolfsburg later on Saturday, on 55 points after snatching a 1-1 draw at Werder Bremen.
Borussia Moenchengladbach also kept up their hopes of a top- four finish when they eased past relegation candidates Nuremberg 2-0 with goals from Juan Arango and Max Kruse to move ahead of Bayer Leverkusen into fourth place on goal difference.
VfB Stuttgart threw themselves a lifeline with five games left when they beat Freiburg 2-0 to climb to 15th place on 27 points, level with Hamburg SV, who beat Bayer Leverkusen 2-1 on Friday.
Leverkusen sacked coach Sami Hyypia earlier on Saturday after going nine games with only one win and slipping out of the top four.
Image: Bayern Munich's Bastian Schweinsteiger reacts during their match against Augsburg.
Photograph: Michael Dalder/Reuters
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