Stefan Kiessling gave Sami Hyypia's men the lead in the 11th minute, but Tobias Werner levelled on the hour mark.
Heung-Min Son put Leverkusen in front again in the 80th minute before Emre Can sealed victory three minutes later.
On a run of nine games without a win, the best thing Leverkusen could have got in Augsburg was an early goal, and thanks to some weak defending, they were granted that wish.
Son sent the ball into the path of Gonzalo Castro down the right wing and his cross found Kiessling in space. The Germany international's first touch was perfect, taking him past the last defender before he slotted his shot in.
Having got the lead they were looking for, Leverkusen then sat back and defended, inviting Augsburg to attack them.
They held firm, frustrating their hosts who could not find a way through before the break.
Indeed, they had to wait until the hour mark before levelling, after increasing the pressure even further.
Werner netted it when he raced in to meet Ronny Philp's cross from the right with a determined header from six yards out, beating a slightly reticent Giulio Donati to the ball.
Leverkusen reacted strongly, though, and pushed for a second goal, which they found with 10 minutes remaining.
Son exchanged passes with Julian Brandy before threading his shot inside Marwin Hitz's post with the Augsburg goalkeeper caught out on his return to the starting XI in place of Alexander Manninger.
Just three minutes later, Can raced through and sent his left-footed shot in off the inside of the far post to seal the win.
Having scored his one previous goal this season against Augsburg, he showed how happy he is to face Markus Weinzierl's men, who slip three points adrift of the top six as a result of this home defeat.