Sami Hyypia's men were forced to come from behind at the BayArena on Saturday, after Dortmund loanee Moritz Leitner had given Stuttgart the lead after 12 minutes.
Stefan Kiessling's 10th Bundesliga goal of the campaign levelled matters 14 minutes later, before Eren Derdiyok struck six minutes from time to make it eight wins from 10 home league games for Leverkusen this term.
The win moves Hyypia's side back to within 10 points of runaway leaders Bayern Munich, who host Eintracht Frankfurt on Sunday.
Meanwhile, fourth-placed Schalke kept up the pressure on Dortmund and Leverkusen by beating Wolfsburg 2-1 at the Veltins-Arena.
Felipe Santana broke the deadlock for the hosts nine minutes in, before teenager Maximilian Arnold brought Wolfsburg level midway through the second half.
However, Kevin-Prince Boateng's 81st-minute winner ensured Schalke made it four home Bundesliga victories in a row.
Hamburg's defeat at Hoffenheim condemned them to a fifth consecutive top-flight reverse for the first time since 1970 as their slump in form continued.
Bert van Marwijk's side lost 3-0 at the Wirsol Rhein Neckar Arena courtesy of goals from Roberto Firmino, Niklas Sule and Andreas Beck, and now sit just two points clear of the automatic relegation zone in 16th.
Borussia Monchengladbach slipped to their second straight league loss as Hannover beat them 3-1 at the HDI Arena, with Mame Biram Diouf scoring twice for the hosts.
Elsewhere, Augsburg made it eight Bundesliga outings without defeat with a 3-1 home win over Werder Bremen, while Mainz overcame struggling Freiburg 2-0.
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