A stunning first half performance saw Sebastian Freis grab the only goal, although Robert Zulj did miss from the spot.
The second period was opened up by Serdar Dursun's strike, though Mario Kvesic and a Marcel Franke own-goal levelled proceedings.
Franke's day would not be ruined, however, as he popped up at the death to head home a last-gasp winner.
Fürth show their attacking intent in the first half
The game began slowly but built up to be an entertaining and attacking affair. A superb pass from Andreas Hofmann made the first chance, as he threaded Veton Berisha through on goal. He did well to bend his run to stay onside, but Berisha couldn't match that with the finish as he shot over.
Aue responded and came close to taking the lead themselves, only to see Pascal Köpke's deflected shot, off Marco Caligiuri, tipped away by Balazs Megyeri. The resulting corner brought another chance for the Violets, but he headed straight at Megyeri.
Sustained pressure followed from Aue as Christian Tiffert fired narrowly wide, with Berisha repeating the feat at the other end. The Norwegian was involved again moments later, this time he was fouled by Steve Breitkreuz; the defender caught Berisha's leg while clearing. Zulj stepped up to the spot, but fired well over the bar.
Bersiha had a header superbly saved by Martin Männel and the rebound was fired wide by Freis in a period of real dominance for the Shamrocks. It didn't look like they would go in ahead at the break, but Freis collected Zulj's pass, worked round Breitkreuz and then poked in at the near post to give his side a deserved lead.
Aue's comeback cancelled out by Franke
The second half picked up where the first left off, and Fürth should have doubled their lead but for Männel to deny Zulj. That mattered little, though, as Dursun took care of that just one minute latter. Khaled Narey's cross was excellent, and left the new signing with a simple finish for his first goal in green and white.
Comfortable in the game, the hosts kept coming forward only to be punished by a superb piece of individual skill. Kvesic picked the ball up just outside the box and weaved his way past several Shamrock defenders before sliding the ball under the oncoming Megyeri to reduce the deficit to 2-1.
The two teams went back and forth but it was Aue who would strike the next blow. Two substitutes combined to create the goal, as Fabio Kaufmann set Cebio Soukou free on the right. His run and resulting effort saw Megyeri save, but only parry the ball straight at Franke and the defender put through his own net to level.
It seemed destined to ended in a draw, with even Tiffert's late red card for two quick yellows seemingly meaning little to either side. However, Narey's crossing came to the fore once more and another pin-point delivery found Franke at the back post to head past Männel and secure all three points in the 94th minute.