Paul Verhaegh's second-half penalty saw Augsburg inflict a seventh defeat of the season on the Bundesliga's bottom club Stuttgart.
The hosts' task was never going to be easy after Daniel Schwaab's first-half red card, and Augsburg's win takes them up into the top six.
Schwaab received two bookings inside the opening half an hour and Augsburg almost made their numerical advantage pay instantly, but Raul Bobadilla's shot was acrobatically saved by Sven Ulreich.
Markus Feulner and Halil Altintop went close after half time as Augsburg continued to press for the opener before they finally broke the deadlock in the 72nd minute.
Visiting skipper Verhaegh made no mistake from 12 yards to take his goal tally for the season to four after Adam Hlousek had handled the ball in the area.
