Andre Hahn put the visitors in front with an eye-catching volley but Leverkusen worked their way back into the game with captain Simon Rolfes levelling and Emre Can grabbing a winner.
Leverkusen remain level on points with second-placed Borussia Dortmund, just a point behind Bayern, but they could fall further behind if Hoffenheim's appeal against last weekend's 2-1 win for Leverkusen is successful next week.
It has been a week in which Stefan Kiessling has done everything with his head, or almost.
After seeing his header go through the side-netting to earn Leverkusen a win against Hoffenheim a week ago, he was on target twice as Leverkusen beat Shakhtar Donetsk in midweek.
This time, he nearly scored an own goal with a flicked header going just wide of his far post.
Augsburg did not need any favours, though, with Hahn showing how capable he can be to score on his own, driving an unstoppable volley past Bernd Leno from Tobias Werner's cross in the 24th minute.
It was a deserved lead for the visitors, who had hit the bar through Jeong-Ho Hong moments earlier.
Leverkusen rallied immediately and found an equaliser in the 34th minute with Rolfes turning the ball in from close range after Sebastian Boenisch headed a corner on for his captain.
Kiessling nearly beat Alexander Manninger early in the second half, but the Austrian goalkeeper made the save before seeing his defence come to his rescue to keep Heung-Min Son's effort on the rebound out.
Leverkusen thought they had finally taken the lead in the 72nd minute when substitute Jens Hegeler kicked the ball out of the hands of Manninger to tee up Kiessling, but the referee said the goalkeeper had full control of the ball and the goal was ruled out.
The home team finally went in front in the 83rd minute when Robbie Kruse sent a cross in from the right and Can headed past Manninger to open his account and earn Leverkusen the points.