Gabon striker Aubameyang cemented his status as the Bundesliga's top scorer with his 16th and 17th league goals in 14 games this season after Gonzalo Castro had headed Borussia into an early lead.
Daniel Didavi pulled a goal back for strugglers Stuttgart with half-time approaching.
But a Georg Niedermeier own-goal left the visitors 3-1 down before Aubameyang grabbed their superb fourth goal and his second to inflict a third straight defeat on Stuttgart.
"That was a tough piece of work," admitted Dortmund coach Thomas Tuchel.
"We didn't play that well in the first half, but we were much better after the break.
"We stopped them having any shots on goal and could have won by more, so all in all it was a deserved win."
Hummels found himself on the bench at Dortmund's Signal Iduna Park in the wake of poor performances.
He scored an own goal in Dortmund's shock 3-1 defeat at Hamburg ten days ago in the Bundesliga, then conceded the penalty which sealed their 1-0 defeat on Thursday at Russia's FK Krasnodar in the Europa League group stages.
The 26-year-old came on for the final 10 minutes against Stuttgart for Germany winger Marco Reus with Dortmund 3-1 up.
Stuttgart's caretaker coach Jurgen Kramny got off to a losing start after replacing Alexander Zorniger in the wake of last weekend's 4-0 thumping at home to Augsburg and the defeat left his side second from bottom.
Third-placed Wolfsburg, who had Brazil defender Dante sent off, are now seven points behind Borussia after their goalless draw at strugglers Augsburg.
Both Schalke and Bayer Leverkusen lost ground behind the league leaders Bayern Munich after taking a point each in their 1-1 draw as Eric Choupo-Moting's goal for Schalke was cancelled out by a late Sascha Riether own goal.
Both clubs are now 19 points behind Pep Guardiola's Bavarian giants.
- Leap-frog -
The runaway league leaders are eight points clear of Dortmund after grinding out a 2-0 win at home to Hertha Berlin on Saturday thanks to goals by Germany forward Thomas Mueller and France Under-21 winger Kingsley Coman.
Bayern show no signs of loosening their iron grip on the Bundesliga title.
They picked up their 13th win from 14 league games to give them 40 from a possible 42 points and this was the first time in eight home games that Bayern failed to score at least four goals.
Mueller, who has 13 goals, has already equalled his tally for the whole of last season with less than half the campaign gone.
Hertha, who last won in Munich in 1977, are in fifth place after Borussia Moechengladbach leap-frogged them into fourth with a 3-3 draw at Hoffenheim.
Gladbach's head coach Andre Schubert preserved his nine-game unbeaten record since taking charge two months ago as USA international Fabian Johnson scored Borussia's late equaliser.
Hamburg are up to sixth after their 3-1 win at Werder Bremen in the north German derby.
Hamburg raced into an early 2-0 lead as Croatia midfielder Ivo Ilicevic scored on their opening attack, then striker Michael Gregoritsch doubled their lead on 26 minutes.
Nigeria's Anthony Ujah pulled a goal back for Werder on 62 minutes, then Germany forward Nicolai Mueller made it 3-1 to Hamburg on 68 minutes when Ilicevic's final pass put him in behind the hosts' defence.
Hanover eased the pressure on coach Michael Frontzeck with a thumping 4-0 victory at home to Ingolstadt which lifted them up to 14th and pulled them away from the relegation zone.
Japan striker Yoshinori Muto scored his seventh goal of the season in Mainz's 2-1 win at Eintracht Frankfurt.
His forward partner Yunus Malli added their second to claim his eighth while Frankfurt had captain Alex Meier sent off for a second yellow card.