Borussia haven't won since mid-January and are without a victory since it was announced Marco Rose will coach Dortmund next season.
They are also facing Champions League misery in Budapest on Tuesday when they will attempt to overturn a 2-0 deficit in the second leg of their last-16 tie against runaway Premier League leaders City.
On Friday, they squandered a chance to take a first-half lead when Lars Stindl missed a penalty.
Substitute Ruben Vargas gave Augsburg a 52nd-minute lead which was cancelled out 15 minutes later by German international Florian Neuhaus.
Marco Richter restored the home team's lead with Andre Hahn adding a third goal in the dying moments.
The win lifted Augsburg five points clear of the relegation zone while Borussia are in 10th spot, 10 points off the Champions League places.