Stuttgart's hopes of avoiding relegation took a hit as goals from Werner and Bobadilla left them empty handed leaving Augsburg.
Raul Bobadilla's 73rd minute strike sent Augsburg to fifth in the league, one point in front of Schalke.
Tobias Werner gave Markus Weinzierl's side the lead very early on, latching on to Alexander Esswein's ball, driving the ball into the bottom corner.
After starting well and reaping the rewards through Werner's goal, Augsburg continued to dominate proceedings. Yet despite looking by far the better team, Stuttgart responded to conceding early.
Daniel Ginczek finished from close range past Marvin Hitz after Filip Kostic provided the opportunity.
An equaliser couldn't stop Augbsurg, however, with Esswein missing from inside the area. It couldn't stop Stuttgart either, spurring them on to attack further forward - Georg Niedermeier got on the end of a cross well but couldn't steer it past Hitz who claimed comfortably.
Hitz had to work again when Serey Die blasted a shot from 20-yards just before half time.
Stuttgart continued to move into the ascendency after the break with Didavi coming close to striking past Hitz from a long way out.
After Stuttgart's run of chances, Augsburg couldn't break down their defence. Long range shots from Verhaegh, Werner and Esswein was all they could muster, and were no trouble for Stuttgart.
It was as they produced an attack of quality that they regained their lead. A low cross into Raul Bobadilla provided a great opportunity and the 27-year-old slammed it high into the top of the net.
Stuttgart pressurised, in hope of saving themselves from dropping further into the relegation scrap and did so well. Christian Gentner wasted a good opportunity and despite more of the ball than before Augsburg's winner, they couldn't press forward far enough and break down the home side.
The visitors had 16 attempts but half were from set pieces, showing their lack of quality in terms of open play. Stuttgart slip into the bottom two as Augsburg slip above Schalke into fifth.
Stuttgart are looking dangerously close to the 2. Bundesliga and have the chance to propel themselves away from that bottom two with a match against Freiburg next season who are level on points with them.
For Augsburg, it is too far fetched to foresee an opportunity to overtake Borrusia Monchengladbach who are 12 points in front. Yet this win takes them above Schalke, who have a game in hand, and more importantly further away from Hoffenheim and Borussia Dortmund who could have been too close for comfort had they not won on Saturday. Europe is in sight for them, 2. Bundesliga is in sight for Stuttgart.