The hosts looked to be on their way to a comfortable win when Christian Gentner and Martin Harnik struck inside 20 minutes, but Reus swiftly reduced the arrears, before completing the turnaround either side of Georg Niedermeier's 67th-minute sending off to send Dortmund back above FC Schalke 04 into second in the table.
Stuttgart at the double
Not helped by the distraction of Wednesday's upcoming UEFA Champions League quarter-final first leg in Madrid, Dortmund found themselves chasing the game inside ten minutes. Oliver Kirch was caught on the ball on the edge of his own box, allowing Stuttgart captain Gentner to steal possession, scuttle past Sokratis and guide a well-measured effort into the bottom corner.
Utility man Kevin Großkreutz forced a smart save out of Stuttgart's Sven Ulreich in response, but there was no disguising Dortmund's uncertainty at the back. Ibrahima Traore led young Erik Durm off on a merry dance down the BVB left and, with Mats Hummels having to step out of position to cover, Harnik was left all alone to sweep in the Guinean's inch-perfect cross from close range.
Reus reduces the arrears
Stuttgart almost made it three with a virtual carbon copy of their second goal, but Traore's final ball left a lot to be desired and Vedad Ibisevic could only find the side netting. Dortmund made the hosts pay two minutes later when Jonas Hoffman slid in Reus for his tenth strike of the season, before Henrikh Mkhitaryan passed up a gilt-edged opportunity to bring his side level barely 60 seconds later.
Dortmund were beginning to get on top of Stuttgart towards the end of the first half as Reus clattered a nailed-on equaliser against the outstretched leg of Niedermeier. The visitors cashed in another one of their get-out-of-jail-free cards after the restart, with the upright coming to Ulreich's rescue to keep out Robert Lewandowski's piledriver.
BVB's leading marksman was making a real nuisance of himself and when Niedermeier sent the Polish goal-getter tumbling in the box, the referee had no option but to point to the spot and produce a red card. Reus stepped up to send Ulreich the wrong way and bring his side back on level terms with just over 20 minutes left on the clock.
An injury to referee Michael Weiner offered Huub Stevens' men only momentary respite as Reus went within a fine Ulreich save of his hat-trick, minutes before sealing the win with the aid of a huge deflection. Klopp could barely maintain his emotions, but his heart must surely have skipped a few beats late on when the unmarked Harnik collected the ball in the box, only to slide his effort mere inches wide of the far post.
Line-ups and statistics
Click below to watch Marco Reus' brilliant hat-trick as he single-handedly dismantled Stuttgart, courtesy of the official Bundesliga YouTube channel: