The result sees Stuttgart go level on 26 points with Leverkusen but they remain in fifth place behind their opponents due to an inferior goal difference.
Even though they were playing away from home, had less possession and created fewer chances, Stuttgart produced more xG than Leverkusen because of the superior cutting edge they showed with the chances they did create and blew their hosts away with a blistering first-half performance.
Stuttgart burst out of the blocks, with the visitors creating a chance for Josha Vagnoman virtually from the kickoff but Mark Flekken made a good save.
However, in the seventh minute, he was beaten when Jamie Leweling rounded him with some deft dribbling before slotting home an easy finish to open the scoring. Leweling thought he had a second moments later but it was ruled out for offside.
They were able to double their advantage with a penalty in the 29th minute after Vagnoman was fouled in the box by Jeanuel Belocian and Maximilian Mittelstadt made no mistake from the spot.
The away side's fortunes got even better on the stroke of half time with a double strike in injury time at the end of the first half to put them 4-0 up at the break.
Deniz Undaz set up Leweling's brace before getting on the scoresheet himself three minutes later to all but secure the result for his side.
Leverkusen were much better after the break and managed to pull one back through an Alejandro Grimaldo penalty in the 66th minute but that was all they could muster as Stuttgart humbled them on the day.
Elsewhere in the Bundesliga on Sunday, Freiburg claimed an important 2-1 win over HSV at Europa Park Stadion.
After a dull first half, Luka Vuskovic put the visitors ahead three minutes after the break but going down a goal proved to be the catalyst Freiburg needed.
Vincenzo Grifo levelled proceedings from the penalty spot in the 53rd minute before Igor Metanovic bagged a late winner in the 83rd minute to claim the victory.
Union Berlin recovered from two goals down to salvage a 2-2 draw with Mainz at An der alten Forsterei.
Nadiem Amiri opened the scoring for the visitors in the 30th minute and Benedict Hollerbach doubled their advantage in the 69th minute.
Woo-Yeong Jeong pulled one back in the 77th minute before a late Murin Ljuobioic goal in the 86th minute rescued a point for the hosts.
Heidenheim and Koln played out to a 2-2 draw at Voith Arena with the majority of the action coming in the first half.
Marvin Pieringer put Hoffenheim ahead in the 15th minute but it was quickly cancelled out by Eric Mertel's leveller three minutes later.
Julian Niehues restored the hosts lead in the 26th minute, with Said El Mala scoring for Koln shortly after half-time to ensure the spoils were shared.
The RB Leipzig vs St Paul and Werder Bremen vs Hoffenheim fixtures were postponed due to unsafe wintery conditions.