Jan Hochscheidt gave the visitors the lead, but Alexandru Maxim and Martin Harnik turned the game on its head to leave Stuttgart ahead going into the final 10 minutes.
They could not hold on for a much-needed win, though, as Ermin Bicakcic equalised eight minutes from time to keep Braunschweig's own hopes of avoiding the drop alive.
After losing eight games in a row, it was only natural that Stuttgart lacked confidence against a side who were putting pressure on them from the off.
Hochscheidt had the ball in the back of the net in the 13th minute, but was fortunate not to be shown a yellow card since the whistle for offside had long since sounded.
He scored a perfectly valid goal midway through the first half, though, to give Braunschweig the lead.
Karim Bellarabi advanced down the left with the ball at his feet and he cut inside before crossing the ball, with the aid of a deflection, into the path of Hochscheidt, who beat Sven Ulreich stylishly.
Stuttgart, who have not managed to win a game after falling 1-0 down so far this season, were up against that record and some already disgruntled fans, but they responded well, equalising within six minutes.
Maxim, whose glaring miss a week ago prompted their collapse against Eintracht Frankfurt, this time scored a more difficult goal from close range, showing more strength than Bicakcic to bundle the ball in.
Five minutes later, Stuttgart took the lead with Maxim this time the provider, crossing for Harnik to take the ball down on his chest before sending his shot into the left-hand corner from outside the area.
Stutttgart could have wrapped up victory in the 52nd minute when Daniel Davari fouled Harnik inside the penalty area.
However, the Braunschweig goalkeeper made amends for his mistake by keeping Christian Gentner's spot-kick out.
For the second week running, Stuttgart missed a glorious opportunity to open up a two-goal lead, and once again, they were punished.
Dominik Kumbela flicked the ball on from a corner to Bicakcic, who made up for his failure to prevent Maxim's goal to score what could prove to be a crucial goal for Braunschweig, and for Schneider.