Hungary contained their rivals and took the lead when Adam Szalai slotted in Laszlo Kleinheiser`s pass.
A disappointing Austria had Aleksandar Dragovic sent off for a second bookable offence before Zoltan Steiber sealed Hungary`s win with a composed dink.
Austria buckle under pressure
Austria had high hopes at Euro 2016 after they won nine out of 10 matches in an unbeaten qualifying campaign.
However, Alaba`s early effort aside, they struggled to live up to the expectation as key forwards Zlatko Junuzovic, Marko Arnautovic and Marc Janko were comfortably marshalled by the Hungary defence.
Arnautovic did produce one lovely flick in a one-two that opened up the Hungarian defence but, after receiving the ball at the far post, Martin Harnik mishit his effort at goal.
Junuzovic, who had 10 assists in the Bundesliga as he helped Werder Bremen avoid relegation last season, went off with an ankle injury and Austria manager Marcel Koller will be concerned by the seriousness of that as well as the lack of cohesion and quality from his side.
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