The forward received his marching orders after an idiotic clash with Koo that left the visitors having to hold on for 45 minutes with a man down in a bad-tempered derby clash
It was not all plain sailing for Jupp Heynckes' men, however, who had to play almost the entire second half with 10 men following a straight red card for Franck Ribery.
Playing for a place in the last-eight of Germany's domestic cup competition, Bayern started brightly and soon settled into their customary rhythm, spraying the ball around with confidence and looking for the gaps to appear in the home defence.
It was Augsburg, however, who threatened first to open the scoring. An excellent passing move culminated at the feet of Ja-Cheol Koo, who hit towards goal from outside the area. A wicked deflection off the heels of Dante threatened to catch out Manuel Neuer, but the goalkeeper responded with a magnificent save to turn the shot onto the post.
As the half progressed, Die Roten began to look more assured around the Augsburg area. Bastian Schweinsteiger went close with a well-hit effort that whistled past the post of Alex Manninger, while Xherdan Shaqiri also worried the goalkeeper with another shot from distance.
And 26 minutes into proceedings, the visitors found an opening to force themselves into the lead.
Shaqiri played a fluid one-two with Toni Kroos, and opened up the game for Ribery, who had found space on the left. The France international feinted to take on his marker, but opted to send in a low cross that was turned in expertly by Mario Gomez to send the Bundesliga leaders 1-0 up.
But Augsburg were not discouraged, and it took another wonderful stop by Neuer to prevent Mulders levelling the encounter after breaking clean through.
The half ended with a flourish, as Bayern looked to press home their advantage. A long-range bomb from Anatoliy Tymoschuk came crashing down off the crossbar, appearing to have crossed the line. There was no whistle, though, and from the resulting rebound Manninger pulled off an inspired reaction stop to deny Schweinsteiger before the midfielder hit his follow-up once more against the woodwork.
Barely a minute after the restart, the game suddenly descended into chaos. Ribery and Koo clashed strongly after the France international took offence at his rival's tackling, and the scuffle peaked with the Bayern man throwing a fist. The South Korean earned just a yellow for his part in the melee, but Ribery was not so fortunate as he saw a straight red card that reduced Bayern to 10 men.
The next few minutes saw rising temperatures on the field, and Tymoshcuk, David Alaba and Dante were all booked for a handful of of uncompromising tackles.
With Bayern rattled and looking for vengeance, the home team took the chance to exploit the numerical advantage. A goal following Werner's free kick was correctly ruled out for offside, before the Augsburg attack were left indignant when a handball from Dante in the area was deemed accidental.
The away team, meanwhile, began to try to take the rhythm out of the game, keeping possession and spoiling Augsburg's attempts to break forward. That is not to say, however, that the league leaders did not have chances. Schweinsteiger was again left ruing his luck when he was played in against Manninger, only for Ragnar Klavan to intervene with a perfect last-ditch tackle.
Substitute Mario Mandzukic also squandered a golden chance to kill the game off in the dying minutes, but it mattered not as Shaqiri ended the match with a smart close-range finish to assure their passage into the next round.