Sascha Molders made it 1-0 for Augsburg in the 61st minute while Marcel de Jong doubled the advantage in the 83rd minute and Ji Dong-won pushed the lead to 3-0 after 86 minutes.
Augsburg had lost three of their last four games but jumped to 30 points with the win - although they remain in the relegation play-off spot.
Stuttgart saw their three-game undefeated streak snapped to remain on 39 points.
The hosts had the first chance in the 24th minute with Ji playing Andre Hahn, whose one-on-one attempt from 14 yards was stopped by Sven Ulreich.
Stuttgart were close six minutes later but Vedad Ibisevic's attempt from a cross by Ibrahima Traore went just wide.
Ibisevic then headed a cross from the right over from six yards in the 39th minute.
The visitors nearly grabbed the lead four minutes after the re-start but Augsburg goalkeeper Alex Manninger made a superb save on Martin Harnik's header.
And it was Augsburg who went ahead after 61 minutes with Molders heading home a Tobias Werner corner from the left.
Ji twice was close to doubling the lead but shot too high in the 63rd minute and 72nd minutes.
Stuttgart nearly equalised in the 77th minute but Daniel Baier headed away Georg Niedermeier's header off the line.
Augsburg got their second goal six minutes later with a superb goal by De Jong, who following a long run down the left looped a shot over Ulreich from 24 yards - just 73 seconds after coming into the game as a substitute.
Ji put the finishing touch on the victory in the 86th minute, flashing a volley past Ulreich from Hahn's cross.