The result propelled Serbia to the top of the section on seven points from three games, ahead of Ireland on goal difference.
The Irish kept up the pace with a 3-1 win at Moldova, while Wales slipped to third on five points after a 1-1 home draw with Georgia. Austria have four points, Georgia one and Moldova none.
The Austrians twice came from behind to cancel out an Aleksandar Mitrovic double but were undone by a 74th-minute volley by Tadic with his weaker right foot.
Roared on by their boisterous fans in Red Star's stadium, Serbia drew first blood in the sixth minute when a delightful back-heel by Filip Kostic released Tadic and he squared the ball to Mitrovic for a simple finish.
A calamitous error by centre back Stefan Mitrovic gifted Marcel Sabitzer a 15th-minute equaliser, as the forward had all the time and space in the world to drill his shot past keeper Vladimir Stojkovic.
Mitrovic, who had not scored in a competitive match for Serbia since a 1-1 draw with Croatia in September 2013, restored the home side's lead midway through the first half with a thumping header after Tadic's inch-perfect cross.
Chances went begging at either end before more sloppy defending by the home side allowed Marc Janko to rip through the middle and beat Stojkovic.
Austria looked more likely to grab a winner but Tadic, who also scored one goal and set up the other two in Serbia's 3-0 win in Moldova on Thursday, delighted the home fans.
The Southampton winger got on the end of a loose ball in the heart of the penalty area and drilled his shot past Ozcan, with Serbia's supporters celebrating by lighting several flares.
Austria coach Marcel Koller heaped praise on Tadic while criticising his team's performance.
"Tadic had an outstanding evening and without a doubt he decided the game," he told a news conference.
"We weren't compact enough in the first half and although we improved in the second, we didn't do enough to get a better result."
Koller's Serbian counterpart Slavoljub Muslin was overjoyed.
"We were up against a high-quality team and the good thing is that we are improving all the time," he said.
(Editing by Martyn Herman)
