Jara, who failed to score a decisive spot-kick against Brazil at the World Cup and saw his country eliminated as a result, again failed to convert from 12 yards as third-tier side Chemnitzer prevailed 5-4 on penalties after a thrilling 5-5 draw.
Chemnitzer had looked dead and buried in the first-round tie when Shinji Okazaki put the visitors 2-0 up in the 49th minute, but that only served to spark the lower league side into life.
Within four minutes they were level as Anton Fink struck twice to stun Kasper Hjulmand's side, yet Mainz still looked like progressing when Koo Ja-cheol restored their lead.
A Niko Bungert own goal made it 3-3 and took the match to extra-time, after which Chemnitzer moved in front only to be pegged back when Bungert found the back of the right net.
As penalties loomed, a dramatic finale saw emotions torn to shreds. Marco Kehl-Gomez gave the hosts the lead on 119 minutes, but a poor clearance from goalkeeper Philipp Pentke in the third minute of injury time allowed Johannes Geis to equalise with an audacious effort from inside his own half.
Jara, a close-season signing for Mainz, was the only man to miss from the spot thereafter as the Bundesliga side were eliminated.
On a night full of goals, Bayer Leverkusen and Stefan Kiessling enjoyed their visit to Alemannia Waldalgesheim, winning 6-0.
Kiessling ran riot, scoring Leverkusen's first five goals before Son Heung-min added a sixth with eight minutes remaining.
Four of Kiessling's strikes came in the opening 41 minutes as Roger Schmidt's men dominated their hosts.
Nuremberg, relegated from the top flight last season, saw their new campaign get off to a bad start as they were beaten 1-0 by Duisburg.
