A comfortable win for the visitors made it eight goals conceded in two games for Stuttgart, who now sit just a point ahead of Augsburg in 16th spot.
It took Augsburg less than 11 minutes to break the deadlock as Raul Bobadilla delivered an inch-perfect through ball into the path of Alexander Esswein, who held his nerve under pressure to slot past the onrushing Odisseas Vlachodimos.
Stuttgart struggled to break up the extended spells of pressure from Augsburg, and the game was effectively over before the break as Jan Ingwer Callsen-Bracker capitalised on poor marking to head home from Daniel Baier's corner.
The onslaught continued as Augsburg pulled further clear nine minutes into the second half when the influential Bobadilla took on two defenders and produced another fine pass to find Koo Ja-Cheol, who fired past a helpless Vlachodimos.
A miserable afternoon for the home fans was almost made even worse late on when Esswein threatened to make the scoreline even more emphatic, but Vlachodimos produced a superb stop as the Stuttgart defence avoided further embarrassment.