Pepe Reina's red card in the early stages gave the visitors the numerical advantage and they were able to capitalise on it with a goal in the 71st minute
Bayern Munich's torrid week continued with a 1-0 defeat at home to Augsburg on Saturday.
Raul Bobadilla grabbed the winner for Augsburg to condemn Bayern to their first consecutive Bundesliga defeats in over a year.
Pepe Reina undermined coach Pep Guardiola's decision to rest players ahead of the Champions League semi-final second leg against Barcelona on Tuesday by getting sent off after just 12 minutes at the Allianz Arena.
The rested Manuel Neuer was hauled off the bench and had a busy afternoon as Augsburg, who missed the penalty that followed Reina's dismissal, battled their way to a rare result against the Bundesliga champions that keeps them on course for Europa League qualification.
Guardiola also rested Rafinha, Mehdi Benatia and Xabi Alonso following his side's 3-0 Champions League capitulation in midweek, with Mitchell Weiser, Dante, and Mario Gotze taking their places.
Bayern's difficulties began after two minutes, when Dante fouled Bobadilla and he took the free-kick himself, firing a thunderbolt from 20 yards that required a good save from Reina at full stretch to push it out to safety.
The hosts looked startled by Augsburg's high tempo start to the game, but Bayern were soon creating chances of their own.
Gotze should have done better after he jinked down the left wing, only to fluff his shot from close range.
He almost made amends with a raking shot from the edge of the box that clipped the post as it went wide with Augsburg goalkeeper Marwin Hitz well beaten.
Disaster struck for Bayern after 12 minutes, when a long ball put Bobadilla through one-on-one with Reina, and the Spaniard clumsily bundled him over inside the box, giving the referee no choice but to send him off.
Neuer's rest from first-team action had been brief, but his first task after coming on for Philipp Lahm was made easier when Paul Verhaegh slammed Augsburg's spot-kick onto the post.
Bobadilla continued to cause problems for Dante in the Bayern defence, and the striker had a claim for a second penalty waved away by referee Sascha Stegemann when the Brazilian struggled to contain him.
A slip from Dante allowed Halil Altintop a clear chance, but he scuffed it wide with the whole goal to aim at.
Bayern spurned a golden chance to take the lead on the brink of half-time, but Robert Lewandowski – still sporting a face mask after fracturing his nose and jaw against former club Borussia Dortmund - fired over the crossbar when sent clean through by Jerome Boateng.
An almost identical chance fell to Lewandowski just after the break, but Hitz was alert to the danger and cleared his lines.
Augsburg had the ball in the net on the hour mark when Hong Jeong-ho scored from a corner, only to see Stegemann rule it out for an infringement on Neuer by substitute Tobias Werner.
Hitz then made a brilliant save to keep the scores level as Lewandowski teed up Weiser for a stinging shot at the other end.
The frailties in Bayern's depleted rearguard were finally exploited after 70 minutes, when substitute Pierre-Emile Hojbjerg broke down the left and played a wonderfully simple pass to Bobadilla, who made no mistake in tucking away his 10th league goal of the season.
The goal prompted Guardiola to send on Rafinha and Claudio Pizarro to try and salvage something from the game, but Augsburg held on for a famous win.