The match started brightly for Dean Smith's side as John McGinn fired the visitors ahead on 20 minutes.
And when Arsenal's Ainsley Maitland-Niles was dismissed in the 41st-minute for a second bookable offence, the balance appeared to tip Villa's way.
However, Nicolas Pepe hauled the Gunners level from the penalty spot on 59 minutes only for Wesley to put Villa 2-1 ahead 90 seconds later with a crisp finish.
But substitute Calum Chambers scored an 81st-minute equaliser before Aubameyang struck the decisive blow with a smart free-kick six minutes from time.
Arsenal move up to fourth place on 11 points while Aston Villa remain in 18th place on four.
Arsenal: 30 Sep v Man Utd (A)
Aston Villa: 28 Sep v Burnley (H)