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.
Augsburg marched ahead in the sixth minute with Altintop converting Paul Verhaegh's low cross from close range after a shot from Sascha Molders had signalled the home side's intent.
It was Augsburg's first goal of the season and they had no intention of relinquishing their lead as they stood stoutly at the back as Stuttgart tried to respond.
The edge of the penalty area was about as far as Labbadia's men were allowed, though, and chances were few and far between as the hosts kept things tight while waiting for chances to break.
One of those came from a set-piece, taken by on loan Stuttgart midfielder Raphael Holzhauser, who delivered a teasing ball for defender Callsen-Bracker to nod in unmarked from close range with Labbadia visibly incensed with his side's defending.
Another dangerous ball from Holzhauser was not dealt with cleanly by Sven Ulreich and Jens Moravek wasted the chance to make it three, shooting the loose ball wide from a tight angle.
Stuttgart were given a lifeline in the 42nd minute when Alexandru Maxim was brought down inside the area by Verhaegh and Ibisevic scored his six goal of the season in all competitions.
Arthur Boka wasted a good chance to bring Stuttgart level early in the second half and his afternoon's work was soon over as Traore replaced him.
The fresh legs set up Ibisevic, who in turn fed Martin Harnik, who ran into a dead end, but the intent was clearly there as the visitors rallied.
That intent was manifested by an over-zealous Traore, who was punished for a foul on Ronny Philp by receiving a harsh red card.
Despite being a man down, Stuttgart committed men forwards and Molders should have finished them off on the counter-attack, but after being denied by Ulreich, he sent the rebound wide of a gaping goal when it seemed easier to score.