After Bosnia striker Vedad Ibisevic headed Stuttgart ahead five minutes before the break, Guinea midfielder Ibrahima Traore's parried long-range shot was netted by Romania's Alexandru Maxim for a 2-0 lead.
The 25-year-old Traore then scored his first league goal of the season when he latched onto Christian Gentner's pass behind the defence to slot home on 76 minutes.
Traore then found acres of space on the left wing and his cross was converted by Austria's Martin Harnik four minutes from time to inflict a sixth defeat from seven games on Braunschweig.
Earlier, Werder Bremen and Nuremberg fought out a six-goal thriller with Dutch winger Eljero Elia scoring twice for the hosts for a 3-3 draw.
Japan midfielder Hiroshi Kiyotake played a key role in sparking Nuremberg's fight back from 2-0 down with a first-half goal before his superb pass saw Czech midfielder Adam Hlousek hit the equaliser.
Bremen raced into a 2-0 lead after Nuremberg defender Berkay Dabanli turned the ball into his own net before Elia bagged his first with a superb shot on 34 minutes.
Nuremberg fought back after Kiyotake produced a goal from nothing just before the break, then Josip Drmic made it 2-2 early in the second half.
Former Juventus star Elia restored Bremen's lead with a thunderbolt shot from outside the area with 66 minutes gone before Kiyotake's through ball saw Hlousek hit the equaliser to share the points.
The result left Bremen joint eighth and Nuremberg in 15th, but still chasing their first win of the season.
On Saturday, Borussia Dortmund consolidated their place at the top with a 5-0 rout of 10-man Freiburg.
Bayern Munich's 1-0 win over VfL Wolfsburg at the Allianz Arena means Pep Guardiola's second-placed side remain level on 19 points with Dortmund, but Jurgen Klopp's team have a superior goal difference.
Dortmund ran riot at Signal Iduna Park with Germany winger Marco Reus and Poland striker Robert Lewandowski scoring twice before Polish midfielder Jakub Blaszczykowski added the fifth 10 minutes from time.
Freiburg's fortunes took a sharp downturn at 1-0 down when Senegalese centre-back Fallou Diagne was sent off after hauling down Lewandowski and Reus drilled home the resulting spot-kick in the 47th minute.
Dortmund host Marseille in the Champions League on Tuesday looking to bounce back from their 2-1 defeat at Napoli while the defeat keeps Freiburg second from bottom.
Ahead of Wednesday's Champions League match at Manchester City, Bayern stuttered to a 1-0 win over Wolfsburg.
"We won, but we must improve a few things," admitted Guardiola ahead of the trip to Manchester and next Saturday's Bundesliga clash at third-placed Bayer Leverkusen.
"I am certain that we will put in a good performance in Manchester and Leverkusen."
Bayern needed an hour to break the deadlock as Germany star Thomas Müller netted his first Bundesliga goal of the season when he met Franck Ribery's pass from the left wing at the far post.
Bayer Leverkusen stayed third, just a point behind Dortmund and Bayern, with a 2-0 win at home to Hanover 96 as Germany midfielders Simon Rolfes and Sidney Sam scored first-half goals.
Leverkusen's Sami Hyypia host Real Sociedad in the Champions League on Wednesday and deserved the win having peppered the Hanover goal with 17 shots compared to the guests' seven.
Ex-Holland boss Bert van Marwijk made his debut as Hamburg coach with a 2-2 draw at Eintracht Frankfurt as the Hanseatic side twice came from behind with Marcell Jansen scoring a late equaliser.
Van Marwijk, who steered Holland to the 2010 World Cup final, saw Hamburg pick up their first point in three matches, but they stay 16th.
Schalke, who are at FC Basel in the Champions League on Tuesday, were held to a thrilling 3-3 draw at Hoffenheim while Hertha Berlin moved up to fifth after coming from behind to beat Mainz 05 3-1 in the capital.