David Alaba's late goal condemned the Irish to defeat in what was a must-win encounter for Giovanni Trapattoni's men after Sweden's victory in Kazakhstan earlier on Tuesday.
The defeat may now signal the end of Giovanni Trapattoni's reign as Republic manager with the Italian expected to pay for the failure to qualify for Brazil next summer.
The win keeps alive Austria's hopes of catching Sweden in second spot but for the Republic, barring victories over leaders Germany and Kazakhstan by sizeable margins in their two remaining games and an unlikely series of results elsewhere, their hopes are finished.
While Austria possessed the greater craft with Bayern Munich schemer Alaba and sidekick Veli Kavlak pulling the strings in the middle of midfield, Ireland's more direct game repeatedly caused problems for the home defence.
On several occasions, front two Robbie Keane and Shane Long got in behind the Austrian rearguard, but could not find the telling ball, and encouraged by their success, Trapattoni asked Jon Walters, who had started on the right side of midfield, to add his brawn to the mix inside the penalty area with Long moving out wide.
Much of the meaningful first-half action was compressed into its final 15 minutes with Anthony Pilkington, making his first start for Ireland, firing into the side-netting 12 minutes before the break.
That was Alaba's cue to take matters into his own hands having played the role of provider for the opening half-hour, and it was he who forced goalkeeper David Forde into an important 35th-minute save with a well-struck left-foot strike from distance.
Forde was called upon once again four minutes later to block Martin Harnik's skidding effort with Austria threatening to make the breakthrough.
However, opposite number Robert Almer was mightily relieved to clutch Long's 40th-minute cross to his chest as he attempted to pick out the unmarked Walters at the far post.
Ireland were dealt a blow within four minutes of the restart when John O'Shea had to leave the field to be replaced by Ciaran Clark.
The visitors appeared to have adopted a shoot on sight policy, although if their intentions were good, their execution was less so with Walters and Long both going for goal from long range, but firing wildly off target.
Full-back Marc Wilson, however, had Almer reaching anxiously towards his top corner with a 55th-minute piledriver which flew just over the angle of bar and post.
Austria boss Marcel Koller, who had replaced Kavlak with Christoph Leitgeb at the break, sent on Marko Arnautovic for Burgstaller on the hour as he attempted to gain a foothold in the game, and the newcomer floated a speculative effort wide of the far post with Forde off his line.
But Andreas Weimann really should have put Austria ahead when he ran on to Christian Fuchs' 66th-minute pull-back, but shot straight at Forde.
Harnik was equally wasteful four minutes later when the Aston Villa midfielder's pass put him in behind Dunne and he lashed the ball high and wide.
Austria were laying siege to the Irish goal and after Paul Green had blocked an Arnautovic header on the line, Seamus Coleman produced two excellent blocks to repel first Julian Baumgartlinger's powerfully-struck effort and then Harnik's follow-up.
But the pressure eventually told with just six minutes remaining when Wilson could only block Fuchs' left-wing cross to Alaba, who gleefully thumped it into the roof of the net.