Reus struck one before the interval and twice after to hand Dortmund a nail-biting victory over their relegation-threatened hosts, who had led through goals from Christian Gentner and Martin Harnik.
Stuttgart's cause was not helped by the dismissal of Georg Niedermeier, having conceded the penalty which led to Reus' second goal.
The visitors' win moved them back a point above Schalke, who had temporarily taken over second place in the Bundesliga standings with a 2-0 triumph over Hertha Berlin on Friday night.
Dortmund were slack in the opening exchanges and handed the Swabians a route to goal as they aimed to pull clear of the relegation play-off position.
Oliver Kirch was dispossessed on the edge of his own box after eight minutes, gifting Gentner the ball in a good area and the midfielder expertly curled past Roman Weidenfeller.
Dortmund had to wait until the 14th minute to get their first sighting of Sven Ulreich bar the pre-match handshakes. Robert Lewandowski had a shot deflected wide, before Kevin Grosskreutz missed from close range.
But Huub Stevens' side were looking sprightly on the break and added a second goal on 22 minutes, Ibrahima Traore leaving Erik Durm for dead and giving Austrian forward Harnik an empty goal to tap home.
Five minutes later, it was another counter-attack that left Dortmund with some problems. Troare's raw pace set up a chance for Ibisevic, but he could only hit the side-netting.
Immediately, at the other end, Dortmund were back in the match. Germany international Reus finished from inside the box after a neat pass by Jonas Hofmann, who was one of the players brought in for this game.
Just passed the half-hour mark and there was more evidence of Dortmund's profligacy. Henrikh Mkhitaryan was the villain in midweek for missing a gilt-edged chance against Schalke - and the same applied here.
The Armenian attacking midfielder looked to place a shot into the opposite corner of Ulreich's goal, but he screwed the shot wide.
Eleven minutes after half-time, Dortmund were unlucky not to level when Lewandowski roamed down the right flank and sent a right-footed effort crashing against the post.
There were still defensive concerns for Jurgen Klopp, with Kirch lucky not to have been shown a second yellow card on more than one occasion.
The utility man was subbed on the hour mark, with Pierre-Emerick Aubameyang coming on.
But the match then hinged on one moment. Lewandowski got the wrong side of Niedermeier and was hauled down by the Stuttgart defender, who was shown a straight red card.
Reus - who only came back to match action 10 days ago after injury - fired home from the spot on 66 minutes to put Dortmund back on level terms.
And the 24-year-old sealed a memorable afternoon with the winning goal seven minutes from time, capping off a fine performance at the Mercedes-Benz Arena.
Aubameyang was the architect, breaking down the left-hand side and cutting back for Reus to slot home his third goal of the afternoon.