Stefan Kiessling opened the scoring for Leverkusen in the 26th minute and Lars Bender doubled their advantage 15 minutes from time.
Sascha Molders made things interesting by pulling a goal back for the visitors in the 89th minute, but they could not force an equaliser.
The result lifted Leverkusen to within 16 points of leaders Bayern Munich, and meant Augsburg remained second-bottom after suffering their first loss since the winter break.
Leverkusen had the first chance of the match after 12 minutes when Augsburg midfielder Jan-Ingwer Callsen-Bracker deflected Daniel Carvajal's cross towards his own goal, but Augsburg goalkeeper Alexander Manninger made a good save.
The hosts did grab the lead 14 minutes later, though. Gonzalo Castro's free-kick found Kiessling, whose header flew in off the right post - his 15th goal of the season.
Augsburg went close in the 34th minute, but Leverkusen goalkeeper Bernd Leno blocked Tobias Werner's shot.
Leverkusen had an opportunity to make it 2-0 a minute later, but Manninger did well to fend away Sidney Sam's shot.
The visitors were within inches of getting back on level terms in the 66th minute through Ji Dong-Won's curling free-kick, which cleared the wall only to drift narrowly wide.
Augsburg missed two further chances in the 74th minute, Leno parrying Michael Parkhurst's blast and Andre Hahn lashing the rebound over the bar.
And moments later Leverkusen had their breathing space. Manninger blocked Andre Schurrle's attempt but the ball fell kindly for Kiessling, who fed Bender to slot home.
To their credit, Augsburg did not give up and halved the deficit a minute from the end when Molders latched onto Hahn's pass and fired home from six yards, but it was not enough to deny Leverkusen the three points.