Toni Sunjic and Kevin Großkreutz grabbed the goals in a very convincing win.
Olaf Janßen made just one change from their win at the weekend, with the injured Tobias Werner replaced by Berkay Özcan.
Torsten Lieberknecht, on the other hand, opted for four alterations to freshen up his side. That meant Onel Hernandez, Joesph Baffo, Mirko Boland and Jan Hochscheidt all came in for Phil Ofosu-Ayeh, Patrick Schönfeld, Salim Khelifi and Christoffer Nyman.
Sunjic edges hosts ahead
It was a tough beginning to proceedings in Stuttgart as two of the league's top teams went toe-to-toe. It was really a sparring period in the opening 20 minutes as the teams aimed to work one another out and take advantage of any space. Emiliano Insua's shot was the only test for either 'keeper early on, but it was simple for Jasmin Fejzic.
As things wore on the Swabians began to take control and took the lead after some very suspect defending from the visitors. Alexandru Maxim's corner was flicked on at the near post by Christian Gentner and met by Sunjic, who headed home from just a few yards out. Four Braunschweig defenders failed to clear, which didn't please Lieberknecht.
Braunschweig were a shadow of the side that have been so dominant in the league this season, with mistakes allowing Takuma Asano to bring a good save out of Fejzic before Saulo Decarli made a vital intervention to cut out his cross before it found Simon Terodde. Just before the break, Domi Kumbela forced Mitchell Langerak into a simple stop.
Großkreutz gives Stuttgart all three points
After a turgid first half, the only solution was to go for three at the back and that immediately gave Braunschweig a shot in the arm. Kumbela couldn't connect cleanly with a cross but Hernandez certainly did with a shot moments later that flew narrowly over from 25 yards. Nyman, who replaced Valsvik to change the formation, also went close.
For all their moment and good play, it was another mistake that gave Stuttgart their second and some necessary breathing space. Asano took advantage of Quirin Moll's misplaced pass and sent a cross to the back post and the unmarked Großkreutz. He controlled expertly before curling into the far corner for his first goal in red and white.
That was really the last major moment in the game, as the hosts strolled to a simple win. They now move just three points behind Braunschweig, who remain top. After the game it was announced that Hannes Wolf would be the new Stuttgart head coach, arriving from Borussia Dortmund's under-19 side.