Leon Goretzka turned into his own net to give the hosts a shock lead after only 13 seconds.
Kingsley Coman restored parity soon afterwards, only for Ji Dong-won to fire Augsburg back in front.
Coman equalised again in first-half stoppage time, before David Alaba completed the comeback.
Goretzka’s opener – the fastest own goal in Bundesliga history – gave Augsburg the perfect start at the WWK Arena, but Niko Kovac’s team soon hit their stride and could have won by a more comfortable margin.
Robert Lewandowski failed to hit the target from a promising position, while Coman slammed a close-range header against the crossbar.
Dortmund can restore their lead at the top to five points with victory over Nurnberg on Monday.
Bayern travel to Liverpool on Tuesday for the Champions League last-16 first-leg tie.