Olic scored from close range with virtually the last kick of the match to hand United their first away loss in the competition this season.
To make matters worse for the English champions, striker Wayne Rooney hobbled off at the end of the match with an ankle problem.
Rooney had got United off to a flying start, volleying them ahead in the second minute, but Franck Ribery's 20-metre free-kick deflected off the wall in the 77th for the equaliser.
In front of a sold-out 66,000 crowd, Bayern were largely kept out of the box by a steely United defence in the first half.
They increased the pressure after the break and were rewarded with the late winner by Croatia international Olic which brought back memories of United's famous comeback victory over Bayern in the 1999 final.
Bayern, without injured Dutch winger Arjen Robben, showed attacking spirit after falling behind to Rooney after only 65 seconds but found themselves unable to threaten the visitors' goal.
Ribery managed to get a shot on target in the 21st minute but too often found himself surrounded by two or three United defenders.
Rooney was a constant threat, with his speed troubling defender Martin Demichelis, playing a full 90-minute game for the first time in more than three weeks after a facial injury, for which he wore a protective mask.
The England striker could have made it 2-0 just before the break when he combined beautifully with Darren Fletcher but his weak left-footed shot was no problem for keeper Hans-Jorg Butt.
Bayern came out fighting in the second half when Olic forced a good save from Van der Sar.
With United seemingly satisfied with their solitary goal, they fell further back as Bayern poured forward.
Ribery gave the hosts hope when he fired a free-kick off the wall to level 13 minutes from time.
United defender Nemanja Vidic rattled the bar with a powerful header in the dying minutes.
But it was Bayern who had the final say deep in stoppage time when Olic picked up a loose ball to wrong-foot keeper Van der Sar with a clever left-foot strike.
NEWS:Ferguson - Poor possession proved costly
NEWS:Ferguson - Rooney injury not serious
NEWS:Van Gaal proud with sweet Bayern victory
WIN:Tickets for second leg at Old Trafford
Follow FFT.com on Twitter
Join FFT.com on Facebook
