Goals from Pierre-Emerick Aubameyang and Ilkay Guendogan, either side of Florian Klein's penalty for Stuttgart, put Juergen Klopp's side in the driving seat.
Germany international Reus, who missed much of the first half of the season through injury but extended his contract with the 2011 and 2012 champions earlier this month, added a third goal late on for the visitors as he scored for the third straight game.
Georg Niedermeier, left completely unmarked, cut the deficit in stoppage time but the hosts ran out of time in the search for an equaliser.
Victory lifted last season's runners-up to 25 points from 22 matches ahead of next week's Champions League last 16 first leg tie with Juventus and plunged 2007 champions Stuttgart deeper into crisis.
"We should not allow the mistakes that led to the two goals we conceded," said midfielder Guendogan. "We can talk about the way we got them but nine points...it can't get better than that."
Gabon international Aubameyang saw an early effort ruled out for offside before he capped an attacking Dortmund start with an easy tap-in following a superb run by Reus in the 25th minute.
Stuttgart bounced back unexpectedly when Niedermeier was brought down and Klein converted the penalty to notch their first home goal of the year.
Dortmund kept up the pressure and reclaimed the lead when a clever flick from Japan international Shinji Kagawa led to Guendogan drilling the ball in.
Reus slotted the third by pouncing on a Timo Baumgartl mistake. The hosts then grabbed their late goal but still remained bottom on 18 points.
Bayern Munich will go 11 points clear at the top if they beat Paderborn on Saturday with second-placed VfL Wolfsburg not in action against Hertha Berlin until Sunday.
(Reporting by Karolos Grohmann, editing by Tony Jimenez)
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