Matthias Ginter had an early chance to score for the hosts, but he could only direct his header wide of the post.
However, within minutes Dortmund took the lead when Aubameyang arrived to tap in Ilkay Gundogan's low ball across the box.
Dortmund continued to look a threat every time they came forward and Marwin Hitz had to be alert to palm Gundogan's curling strike over the bar.
They would not be denied a third goal in the 33rd minute as another flowing team move ended with Reus slotting in from close range, after being set up by Kagawa's backheel.
Augsburg looked in danger of leaking even more goals, but they gave themselves a lifeline four minutes into the second half when Raul Bobadilla met Tobias Werner's free kick and looped a header into the corner to get his team on the scoresheet.
Bobadilla had a chance to add to his tally just past the hour mark, but after finding space inside the box, he failed to hit the target with his effort.
Despite the improvement from the visitors, Dortmund still looked a threat going forward and Aubameyang forced Hitz to beat away his stinging drive.
With five minutes left to play the hosts restored their three-goal lead as Aubameyang slid in at the back post to convert Adrian Ramos's cross and in stoppage time, the striker fired in his 20th goal in his last 17 games to complete his hat-trick and put the seal on a resounding win.