Halil Altintop's first Bundesliga goal since 2010 gave Augsburg the lead and Jan-Ingwer Callsen-Bracker doubled their advantage before Vedad Ibisevic pulled a goal back for Stuttgart before the interval.
Try as they might, Stuttgart were unable to find an equaliser and they had to play the final 20 minutes with 10 men after Ibrahima Traore saw red.
Eintracht Frankfurt beat Eintracht Braunschweig 2-0 to get off the mark with their first points of the season.
After laying siege to their hosts' goal without scoring in the first half, Frankfurt finally went ahead through Alexander Meier in the 52nd minute.
Ten minutes later, Stefan Aigner sealed the first win of the season for Armin Veh's side as Braunschweig suffered a third defeat in as many top-flight games following promotion.