Austria created the first promising chance of the game when Andreas Wiemann got down the right before curling a ball towards the far post, but John O'Shea did well to put the lingering Guido Burgstaller off as the ball bounced behind to safety.
Republic of Ireland hit back soon after to produce the game's first real sniff at goal as Jonathan Walters made the most of an Alaba mistake, allowing him to put a cross to the far post, where Shane Long was just unable to reach it.
Robbie Keane then went close as he drilled an inviting ball on the half-volley across goal, missing both the bottom corner and the advancing Long.
Austria were dangerous on the counter and showed promise when Alaba worked his way to the edge of the area before testing David Forde, who turned the ball away for a corner. Forde was again called into action when Martin Harnik picked up a loose ball before forcing another good save from the Republic of Ireland keeper.
Following the interval Austria spurned a huge opportunity to break the deadlock when Christian Fuchs put in a low cross for Wiemann, who was only able to hit his shot at Forde. They then wasted again when Harnik found space inside the box, but he smashed his effort over.
Austria finally broke the deadlock when Alaba made use of a loose ball with a quick touch, allowing him to crash the ball into the top corner of Forde's goal.