Mainz and their third-tier opponents were level at 3-3 after 90 minutes, and then 5-5 following extra time before Chemnitzer won the shootout 5-3.
Christian Heidel's side, who finished seventh in the German top flight last season, opened up a 1-0 half-time lead courtesy of Niki Zimling's curling shot into the top corner from the edge of the box.
Three goals in the eight minutes immediately after the restart set the tone for the madness that followed as Mainz went two ahead through Japanese international striker Shinji Okazaki, before they were pegged back by an Anton Fink double.
The hosting minnows appeared to have snatched a late winner when another Garbuschewski delivery found Marco Kelh-Gomez to bury from inside the box, but there was to be another twist in the tale.
In the 122nd minute, Mainz equalised through a 30-yard Johannes Geis strike, and Chemnitzer might have been feeling that their chance had gone as the visitors took the momentum into penalties.
Chemnitzer scored all five of their efforts from 12 yards to secure dramatic passage into the last 16 of the competition.