Bruno Labbadia was sacked as Stuttgart boss following their third-consecutive Bundesliga defeat.
Halil Altintop and Jan Callsen-Bracker scored early for Augsburg, but Bosnian Vedad Ibisevic pulled a goal back just before half-time.
Stuttgart were reduced to 10 men when Guinea international Ibrahima Traore was sent off in the 70th minute, just after coming off the bench.
And they were unable to score a late equaliser, forcing the club to axe Labbadia after just three matches of the new campaign.
Meanwhile, Eintracht Frankfurt bounced back from consecutive Bundesliga losses with a 2-0 success against Eintracht Braunschweig on Sunday.
Armin Veh's side, who finished sixth in the German top flight last term, were thrashed 6-1 by Hertha Berlin on the opening day of the season.
That was followed by a 1-0 home defeat to Bayern Munich but Frankfurt, who won in the first leg of their Europa League play-off against Qarabag on Thursday, claimed their first league triumph of the season in Braunschweig.
After a goalless first half, midfielder Alexander Meier broke the deadlock in the 52nd minute.
Meier, who was the third highest scorer in the Bundesliga last season, has already hit four goals in five matches in all competitions for his side in the current campaign.
Frankfurt's win was sealed in the 63rd minute when Stefan Aigner netted as the struggles for the newly promoted Braunschweig continued.
Braunschweig have now lost all of their three Bundesliga matches this season.