Tobias Werner scored the only goal in the 29th minute although Bremen will point to a lack of good fortune after twice striking the woodwork.
Augsburg have now collected 12 points since the winter break - the third-best return in the league behind only leaders Bayern Munich and Borussia Dortmund - but remain cut adrift in the relegation zone.
Bremen, meanwhile, are now nursing a record of three straight defeats and must improve quickly if they are to reignite their bid to qualify for Europe.
Bremen had the first chance after 18 minutes as Marko Arnautovic's run and left-footed shot was kept at bay by Augsburg goalkeeper Mohamed Amsif.
Augsburg grabbed the lead after 29 minutes with Andre Hahn's cross from the right side of the area picking out Werner who headed in from 11 yards.
Bremen were close to equalising after 37 minutes but Nils Petersen could not converted Arnautovic's cross.
Home goalkeeper Sebastian Mielitz made a superb save from Hahn's strike in the 57th minute, and Bremen were unlucky not to equalise in the 68th minute as Kevin De Bruyne's cross from the left found Aaron Hunt who struck against the crossbar from six yards.
Amsif made another great save in the 83rd minute, batting Sokratis' effort over the bar.
And Bremen knew their luck was out when Sokratis fired a shot off the base of the right post in the 87th minute.