The result ensured Dortmund strengthed the credentials for a second-placed finish, with Klopp's men now one point ahead of rivals Schalke, and ensured the hosts remain in the mire.
The hosts took a deserved lead within 10 minutes, when Christian Gentner took advantage of a defensive error to curl home, before Martin Harnik doubled their advantage by rounding off a great run from Ibrahima Traore.
The visitors pulled a goal back 15 minutes before the break through Reus' cool finish, and were nearly level just after the interval when Robert Lewandowski's strike came crashing off the post.
Stuttgart were then reduced to 10 men midway through the second half, when Georg Niedermeier brought down Lewandowski in the area and Reus converting the resulting spot-kick.
Reus was then on hand to complete the comeback with a well-worked third, as his deflected finish from the edge of the area found the back of the net.
Klopp dismissed accusations that his side were distracted by their UEFA Champions League exploits after being held by Schalke last week, but they started slowly against a spirited Stuttgart side.
And they were made to pay for a cagey opening after 10 minutes, when Gentner pounced on Oliver Kirch's poor back pass on the edge of the area, before curling a smart finish past Roman Weidenfeller.
The goal gave Dortmund a little more urgency, with Kevin Grosskreutz forcing a good save from Sven Ulreich five minutes later.
But the home side’s lead was extended after 19 minutes, when Traore powered through on the right hand side before cutting inside to Harnik, who was left with a simple finish from close range.
The visitors pulled one back on the half-hour mark, as a perfectly weighted ball from Jonas Hofmann unlocked the home defence to find Reus, who finished with aplomb.
Henrikh Mkhitaryan nearly levelled just a minute later, finding space on the left before firing a drive just wide of the post.
Dortmund's leading scorer Lewandowski came closer 10 minutes after the break, bursting through the heart of the defence before crashing a vicious strike off the post as Dortmund got on the front foot.
The Poland striker was again at the centre of proceedings 10 minutes later, breaking through on goal before being clipped by Niedermeier in the box, leaving Michael Weiner to produce a red card and point to spot.
Reus made no mistake from 12 yards out and a memorable afternoon for the midfielder was complete seven minutes from time, when he picked up a pass from substitute Pierre-Emerick Aubameyang on the edge of the box before sending a deflected effort past Ulreich.
