Raffael struck from a tight angle late after 37 minutes for his sixth goal of the campaign and their Sweden international Wendt doubled the lead with a stunning shot after controlling a high Patrick Herrmann cross in the 73rd minute.
Coach Lucien Favre's team should have added more goals late on with Germany international Max Kruse twice denied by Stuttgart goalkeeper Sven Ulreich and Herrmann volleying just wide six minutes from time.
It was only Gladbach's third win in their last 31 games against Stuttgart and moved them on to 25 points from 13 matches. They are three points behind Bayer Leverkusen and four clear of VfL Wolfsburg, with both teams playing on Saturday.
Stuttgart, who have now won only one of their last six games, stayed in eighth place on 16.
Leaders Bayern Munich, on 32, visit second-placed Borussia Dortmund, who are four points off the pace, in the big clash of the weekend on Saturday, Bayern are looking for their first league win over rivals Dortmund since 2010.
- Sports & Recreation