Die Roten met heavy resistance from Jos Luhukay's side, who despite fighting bravely for most of the match, were ultimately defeated by their illustrious opponents
Jupp Heynckes' men hit the hosts with a double salvo in the first half through Mario Gomez and Franck Ribery, but Jos Luhukay's men answered with a goal by Hajime Hosogai after the break.
The result sees Bayern Munich increase their lead over Borussia Dortmund to five points before the two teams meet at the Allianz Arena in the next round of the Bundesliga.
The visitors started the game unconvincingly, allowing Augsburg room to operate and attack in the early minutes. The first real chance for the match was for the hosts, who were showing great determination and desire right from the beginning.
In the ninth minute, Sascha Molders reacted quickly to steal the ball off Anatoliy Tymoshchuk on the edge of the box, but Belgian defender Daniel Van Buyten stood strong to thwart the striker's finish.
Bayern Munich soon found their stride and opened the scoring with a quarter of an hour on the clock. A corner on the left by Ribery was met by the head of Van Buyten, whose finish was blocked into the path of Gomez, who easily powered a header home from a yard out.
Heynckes' side continued to press and doubled their lead on the half-hour mark through an immense solo effort by Ribery. The Frenchman cut in from the left, deftly dribbled past Jan-Ingwer Callsen-Bracker and Dominik Reinhardt before slotting past Mohamed Amsif from a tight angle.
The second half saw Augsburg double their efforts as they tried to make the best of Bayern Munich's glaring complacency. Near the hour mark Augsburg finally breached Die Roten's defence through Hosogai.
The Japanese took advantage of the visitors' inability to clear a corner from Tobias Werner from the box, and slammed past Manuel Neuer with a low shot, which David Alaba almost managed to clear off the line.
Smelling blood, Augsburg pushed hard to score an equaliser and spring a surprise against the increasingly uncomfortable Bayern Munich.
Neuer produced a world-class stop in the 83rd minute to deny Edmond Kapllani, but Jupp Heynckes' men, who were reduced to 10 men in the final moments of the match after Tymoshchuk was sent off, ultimately managed to preserve their lead and bag the three points.