Philipp Lahm put Bayern ahead in the second minute and Julian Green headed his first goal for the club.
Bayern keeper Manuel Neuer saved a Koo Ja-cheol penalty, with Thomas Muller, who set up Green's goal, then blazing a spot-kick over for the hosts.
Ji Dong-won gave Augsburg hope with a great finish before David Alaba struck from a tight angle in stoppage time.
Bayern defender Holger Badstuber made his first appearance since February following injury.
German Cup holders Bayern Munich beat Augsburg in a Bavarian derby to reach the last 16.