Tobias Werner, Stephan Hain and Daniel Baier scored for the hosts who were helped by a some woeful Bremen defending while Belgian Kevin de Bruyne replied with his third goal this season.
The visiting strikeforce of Eljero Elia, signed from Juventus in the close season, and Marko Arnautovic were still left waiting for their first goal of the season after another frustrating evening for Thomas Schaaf's side.
Augsburg, who began with only two points and two goals to their credit, got off to a flying start when Paul Verhaegh floated over a long cross from the right and Werner escaped his marker to score with a diving header after two minutes.
However, Bremen had more of the play and equalised in the 19th minute when De Bruyne floated over a free kick from about 30 metres, the ball eluded the defence and bounced into the net.
Augsburg regained the lead 13 minutes later when Hain played a one-two with Knowledge Musona, broke free and squeezed his shot between keeper Sebastian Mielitz and the near post.
The hosts took control of the match in the second half and Zimbabwe midfielder Musona fired wide after the ball rebounded to him in the penalty area.
Baier added the third goal with a low 25-metre free kick which left Mielitz wrong-footed, a first Bundesliga goal for the 28-year-old who has spent most of his career in the lower divisions.
Augsburg now have five points from seven games, only two less than floundering Bremen who have won twice and share the worst defensive record in the league with 12 goals conceded.