Neither side truly stood out in an even contest, but a hapless defensive error with 22 minutes to go allowed Ali Ahmed Mabkhout to score the winner and set up a semi-final with Qatar.
Although the Socceroos made a lively start, UAE improved as the first half progressed and ultimately crafted the better chances.
UAE's productivity in the final third had little to do with what proved to be the winning goal, however, as Degenek gifted it the chance and Mabkhout took full advantage.
It was a particularly sweet win for UAE, defeated by Australia in the semi-finals four years ago.
Australia almost made a flying start, but Trent Sainsbury headed over from eight yards despite being unmarked in the fourth minute.
UAE went close to capitalising halfway through the first period, only for Mat Ryan to make an important block to deny Ismail Al Hamadi after a darting run into the box.
Australia had an even luckier escape on the stroke of half-time – Mabkhout sending a powerful header just over the bar when he looked certain to score.
The Socceroos upped the ante after the interval and went close a couple of times – the lively Chris Ikonomidis shooting wide from Aziz Behich's cross just after the hour.
UAE took advantage a few moments later, as Mabkhout pounced on Degenek's feeble back-pass, skipping past Ryan and slotting into the empty net.
The game stretched into the 101st minute due to several UAE players going down with injuries, but Australia could not use that to its benefit, as Alberto Zaccheroni's men held on.