Shinji Kagawa and Christian Pulisic struck before the break to set them up for a comfortable second half.
Henrikh Mkhitaryan found the net just after half-time, killing the game off as a contest.
Pulisic and Kagawa strike to give BVB half-time lead
Aside from a Stuttgart penalty appeal in the opening exchanges, the game was comfortably controlled by the visitors. Marco Reus had a huge chance to open the scoring when Lukasz Piszczek's shot managed to find the winger, though he couldn't find a finish. Adrian Ramos could have opened the scoring moments later, only to see his shot dealt with by Przemysław Tytoń.
Dortmund continued to pile on the pressure and were rewarded midway through the half. Mkhitaryan, who has been arguably the best player for BVB this season, danced down the left before cutting the ball back into the box. Kagawa was arriving late at the back post, and he had the simple task of tapping into an open goal.
Despite Stuttgart's best efforts to come forward, Roman Bürki remained untested in the first-half. Tytoń was the opposite, and made a good stop to keep out Ramos once more. That would prove in vain, however, as the visitors added a second in the lead up to half-time. A mistake from the Polish stopper saw him push out Mkhitaryan's skidding drive to Pulisic, scoring his second Bundesliga goal in as many games.
Mkhitaryan makes it three
It was a similar story after the restart as Stuttgart pressure was broken by a BVB goal. Mkhitaryan finally got a deserved goal of his own when Kagawa's run and cross was headed goalwards by Ramos and saved by Tyton. Like the second goal, the visitors' were quickest to the rebound and Mkhitaryan was able to pounce to tap home.
That strike killed the tempo in the game, with a Stuttgart double substitution having little effect on the game. Matthias Ginter made a fine block from Filip Kostic and Bürki did well to deny Georg Niedermeier, though the result was already sealed. More changes and missed chances followed, including Moritz Leitner smacking the post. BVB march on and welcome VfL Wolfsburg to the Westfalenstadion next week after keeping their title hopes alive. Stuttgart, on the other hand, face Werder Bremen in a must-win clash.