It took Thomas Schneider’s side 40 minutes to take the lead at the Eintracht Stadion, but there was no stopping them once they had broken the deadlock.
Vedad Ibisevic headed in his sixth goal of the season after an excellent free-kick delivery from Alexandru Maxim, with the Romanian attacking midfielder picking up his fourth assist of the campaign.
Maxim then got on the scoresheet himself five minutes after the break. Ibrahima Traore saw his effort from outside the area palmed away by goalkeeper Marjan Petkovic, but the 23-year-old was on hand to knock in the rebound.
Traore got in on the act in the 76th minute, however. The Guinean attacker played a one-two with Christian Gentner before bursting into the area and slotting past Petkovic.
And Stuttgart completed their scoring four minutes from the end as Martin Harnik tapped in Traore's cross, thus extending Schneider’s unbeaten record since taking charge to four games and leaving newly promoted Braunschweig with just one point this term.
Sunday’s only other match saw even more goals as Werder Bremen were held to a thrilling 3-3 draw at home to Nuremburg.
The hosts were two up by the 34th minute as Berkay Dabanli put Santiago Garcia’s cross past his own goalkeeper, before Eljero Elia connected with Aaron Hunt’s pass to notch his first goal of the season.
But Japanese playmaker Hiroshi Kiyotake pulled one back for the visitors one minute before half-time with an excellent half volley inside the area, before Josip Drmic brought the visitors level eight minutes after the break as he squeezed Tomas Pekhart’s cross past Sebastian Mielitz.
Elia then struck again with a brilliant half volley of his own to put Bremen back in front, but Nuremberg were not to be denied a draw.
Adam Hlousek raced on to Kiyotake’s throughball and fired beyond Mielitz 20 minutes from time to earn a point that moves his side out of the bottom three.
