Leverkusen were on the front foot from the first whistle and were rewarded with the opening goal in the eighth minute when Josip Drmic tapped home after Hitz had saved his first effort.
Karim Bellarabi was a key reason behind his side's dominance early on, and he came close to doubling their lead with a low strike which drifted just wide, after the Germany international had evaded a couple of challenges on the edge of the box.
Leverkusen should have been further ahead before the break, but they missed good chances to extend their advantage, with Bellarabi hitting a tame effort straight at Hitz before Drmic fired wide after latching onto a loose ball in the box.
The impressive Bellarabi tested Augsburg with another run forward early in the second half, but the end product was disappointing as Hitz pushed a shot to safety.
Leverkusen's inability to add to their lead was punished just before the hour mark when Augsburg levelled through Francisco Caiuby, who used the space gifted to him in the box to fire past a helpless Bernd Leno.
Augsburg grew in confidence after finding the equaliser and twice went close to taking the lead as Raul Bobadilla tested Leno with a low strike before somehow managing to slice wide from close range a few minutes later.
It looked as though Leverkusen had clinched all three points with five minutes of normal time remaining. They kept the pressure on from a corner before working the ball to Reinartz, who kept his composure to pick out the bottom corner.
However, Augsburg piled the pressure on in stoppage time, and Hitz, who had been sent forward for a corner, stayed in the box to fire home from close range after Leverkusen had failed to clear a cross.
There was almost one final twist as Halil Altintop rattled the post to come agonisingly close to snatching the win for Augsburg in an enthralling game.