There was an edgy start to the game with all the action taking place in the middle of the pitch. Neither side had a genuine effort on goal in the opening 15 minutes.
The pace picked up slightly and halfway through the half the hosts took the lead. Bungert reacted well to a corner to guide the ball into the far corner with a neat flick.
By now the home side were on top of proceedings and could have gone ahead on 35 minutes, but Maxim Choupo-Moting guided an effort wide of the post.
Shortly after they had the second goal that their play deserved, but there was a huge slice of luck involved. Hitz initially made a decent stop on to the post, but saw the attempted clearance strike him in the face before crossing the line.
Augsburg looked a little brighter after the break, but were still struggling to create as Raul Bobadilla struck a free kick from around 25 yards over the bar.
The away side were never able to get back in the game and instead Mainz made certain of the points in the second half with a goal from Geis, who finished powerfully from outside the box.