Halil Altintop and Jan Callsen-Bracker scored early for the home side, before Bosnian Vedad Ibisevic pulled a goal back just before half-time.
But Stuttgart were reduced to 10 men when Guinea international Ibrahima Traore was sent off in the 70th minute, minutes after coming on as a substitute, as the pressure continues to increase on coach Bruno Labbadia.
One side who enjoyed a positive weekend were Frankfurt, who were thrashed 6-1 by Hertha Berlin on the opening day of the season before losing 1-0 to Bayern, on the back of a sixth-placed finish last term.
But after winning in the first leg of their Uefa Europa League play-off against Qarabag on Thursday, they claimed their first league triumph of the season in Braunschweig.
After a goalless first half, midfielder Alexander Meier broke the deadlock in the 52nd minute. The striker, who was the third highest scorer in the Bundesliga last season, has already hit four goals in five matches in all competitions for his side in the current campaign.
Frankfurt's win was sealed in the 63rd minute when Stefan Aigner netted to pile more misery on the newly-promoted Braunschweig, who have lost all three of their opening Bundesliga games.