Ja-Cheol Koo's reactions proved pivotal in the first-half, as he turned in two shots off the post to give the hosts a comfortable lead.
A stunning third from Koo had seemingly sealed all three points, but Karim Bellarabi responded three minutes later.
Paul Verhaegh's goal and then a last-gasp Hakan Calhanoglu penalty meant that Leverkusen - somehow - took a point away from the WWK Arena.
Koo quickest to react
Augsburg were keen to take advantage of Leverkusen's injury troubles and did just that inside the opening minutes. Another mistake by Wendell allowed Alexander Esswein to steal in front of him and retrieve the ball, before the attacker cut inside and curled an effort onto the underside of the bar. It looked like the chance had gone but the lively Ja-Cheol Koo slammed home the rebound, under Bernd Leno, to give his side an early lead.
The visitors attempted to force a way back into the game immediately, with Hakan Calhanoglu the chief architect of any possible comeback. The Turkish midfielder fired just wide from range before Marwin Hitz held another powerful drive. He didn't come as close as Admir Mehmedi, however. After Karim Bellarabi's promising run, Julian Brandt picked up the pieces and set the ball for the Swiss striker, whose curling effort rattled the post.
With Wendell going close with a free-kick and Bellarabi bringing an excellent stop out of Hitz, it seemed like they had finally found a foothold in the game. However, Augsburg had other ideas. Some excellent work by Alfred Finnbogason saw him skip past Leno but hit the post from an acute angle; the home fans had little to worry about, though, as Koo came steaming in to slot home and give his side a comfortable lead at the break.
Koo completes hat-trick in vain
Christopher Janker was brought off the bench at half-time to replace Jeong-Ho Hong, while Leverkusen remained unchanged. Augsburg continued where they left off before the break, and Esswein came closer still to a deserved goal as he placed a shot just past the post from inside the area.
It would not be long until Augsburg did find a third, however, as Koo completed his hat-trick. A superb move gave the hosts breathing space, with Philipp Max's deep cross excellently chested down to the South Korean by Esswein. Koo swiveled and sent a superb half-volley straight into the top corner, leaving Leno helpless.
Leverkusen, like their midweek defeat to Werder Bremen, found a way back into the game at 3-0 down. Following a speedy break, the ball eventually found Bellarabi in space after Mehmedi's lay-off. The winger stroked the ball home with aplomb, as Augsburg's imperious position looked a little less stable.
The time was ticking by without the home side breaking sweat, but with 10 minutes to go things became interesting. A corner was whipped in hard to the near post area by Calhanoglu and missed by the initial body of players. Paul Verhaegh was caught totally off-guard as a result, and the ball cannoned into the corner.
Hope faded quickly and two minutes of added time looked far from promising, but a dramatic finale ensured Leverkusen would have one final chance. After two shots were cleared off the line, a third was blocked by Jeffrey Gouweleeuw. The Dutch defender used his hands, however, and he was dismissed with a penalty given to boot. Calhanoglu stepped up, stayed calm and sent Hitz the wrong way. The final whistle sounded as the ball hit the net; a breathless game with a finish to match.