The visitors had the game's first chance after just three minutes when Ji Dong-Won saw his effort from outside the box saved.
Augsburg eventually took the lead on 29 minutes through Tobias Werner, who headed home Andre Hahn's pass.
The visitors were the brighter of the two sides in the opening minutes of the second half, Werner and Hahn both trying their luck but in vain.
Bremen were unlucky not to pull level on 68 minutes when Aaron Hunt rattled the visitors' crossbar from close range after receiving Lukas Schmitz's cross.
The home side were denied by the frame of the Augsburg goal again on 87 minutes, this time Sokratis Papastathopoulos being at the receiving end.