Stuttgart have taken 10 points from four matches since Schneider, previously in charge of their under-17 side, replaced Bruno Labbadia, who was sacked after the team lost their first three games of the season.
Vedad Ibisevic broke the deadlock when he headed in Alexandru Maxim's cross five minutes before halftime.
Maxim scored the second five minutes after the re-start, Ibrahima Traore, whose pace on the right caused Braunschweig problems throughout the game, added the third on 76 minutes and set up the fourth for substitute Martin Harnik four minutes from time.
The win lifted Stuttgart to sixth with 10 points from seven games, while Braunschweig are bottom with two and only three goals scored.
Nuremberg came back from 2-0 and 3-2 down to force the second 3-3 draw of the weekend at Werder Bremen in Sunday's other game.
A Berkay Dabanli own goal after eight minutes and Eljero Elia's superb first-time finish put Bremen 2-0 ahead in just over half an hour before Hiroshi Kiyatoke pulled one back just before the break.
Josip Drmic turned in Tomas Pekhart's low cross to equalise in the 53rd minute but Bremen regained the lead with another Elia goal, the Dutchman firing in a low shot from 25 metres 13 minutes later.
Adam Hlousek levelled again in the 70th minute with a delicate finish to Kiyotake's through ball.
Hoffenheim and Schalke 04 also drew 3-3 on Saturday, with the hosts coming from two goals behind.
Nuremberg have drawn five of their seven games and have five points while Bremen are on 10.