Dortmund's 10th win in 14 games tightened their hold on second place, eight points behind leaders Bayern Munich, who beat Hertha Berlin 2-0 on Saturday, and eight ahead of third-placed VfL Wolfsburg, in action at Augsburg later on Sunday.
With out-of-form captain Mats Hummels on the bench, Dortmund got off to a dream start with Gonzalo Castro nodding in on the rebound after three minutes and setting up Aubameyang for his 16th goal of the season in the 19th minute.
Daniel Didavi snatched a 40th-minute goal against the run of play for the visitors, who had interim coach Juergen Kramny on the bench after the departure of Alexander Zorniger this week.
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Dortmund quickly re-established control after the break with Aubameyang, the Bundesliga's leading scorer, coming close to a second goal in the 55th when he fired narrowly over the bar at the far post.
Dortmund, who lost to Hamburg SV in the league on Nov. 20 and FK Krasnodar in the Europa League in midweek, thought they had got it when Marco Reus charged through to put the ball in the net but his effort was ruled offside.
The Germany international did not have to do anything a few minutes later when Stuttgart defender Georg Niedermeier, trying to clear his cutback in the box, slid in for an own goal in the 65th minute.
It was Stuttgart's fourth own goal in 14 league games this season -- a Bundesliga record -- as they remained in the relegation playoff spot on 10 points, having conceded a club record 35 goals so far.
Gabon international Aubameyang fired his 17th goal of the campaign in the last minute to complete their dominant victory in front of an 81,000 crowd.
(Reporting by Karolos Grohmann; editing by Rob Hodgetts)
