Halil Altintop had the first chance of the match but his shot from 12 yards went over the Leverkusen crossbar.
Augsburg were gifted the lead in the 12th minute when a simple back-pass from Jonathan Tah was played to Leno, whose kick skimmed the top of the ball which trickled into the back of the net.
The scores were level in the 39th minute when Karim Bellarabi finished from the centre of the box.
Lars Bender had two shots saved at the start of the second half before Hitz was equal to Kampl's effort as Leverkusen pushed for the lead.
The hosts won a penalty with 20 minutes remaining when Jan-Ingwer Callsen-Bracker handled the ball in the area, but Hakan Calhanoglu fired the ball over the bar as his side were forced to settle for a point.