Arkadiusz Milik put the visitors in front after an even first half an hour and Andre Hahn added a second before the break.
Vedad Ibisevic was sent off for violent conduct early in the second half before Hahn added a third for the guests.
Konstantin Rausch netted a consolation before Tobias Werner completed the rout for the Bavarians, who return to within two points of the top six.
Augsburg did all they could to help Stuttgart early on with Jan-Ingwer Callsen-Bracker three times inviting Ibisevic and Timo Werner to come at them.
Stuttgart did not take their gifts and Augsburg soon stopped dishing them out, instead doling out punishment.
Milik gave the visitors the lead 10 minutes before half time, turning in Werner's cross from close range, and there was only one team in it from that moment.
Werner again went close when he should have served Milik before Hahn doubled their lead just before half time as they accepted another gift from the Stuttgart defence with Halil Altintop sending Hahn through to beat Sven Ulreich.
Stuttgart's frustration boiled over early in the second half when Ibisevic was sent off for a clash off the ball with Callsen-Bracker.
It was not long before Augsburg made the most of their numerical advantage with Hahn making it three in the 56th minute.
He held his nerve once again in a one-on-one situation with Ulreich to put the result beyond doubt and prompt a chorus of jeers from the home fans.
Rausch placated them somewhat, making up for his mistake which led to the second Augsburg goal by grabbing a consolation by taking the ball round Alexander Manninger and slotting it into the unguarded goal, but Augsburg's three-goal lead was restored only two minutes later.
Werner beat the luckless Antonio Rudiger before picking his spot beyond Ulreich and completing a miserable afternoon for the Stuttgart custodian.