Ukraine, who begin their campaign against Sweden before also taking on France and England in Group D, had been looking to pick up a confidence-boosting win in Innsbruck.
But the visitors were caught cold after three minutes when Zlatko Junuzovic curled an excellent free kick beyond the goalkeeper.
Ukraine levelled after 56 minutes, Oleg Gusev tapping the ball in from close range following a through ball from Andriy Yarmolenko.
Marko Arnautovic restored Austria's lead six minutes later with a powerful effort low into the corner.
Gusev, though, struck again in the 65th minute with a long-range thunderbolt of a shot into the top corner.
Arnautovic then grabbed the winner with two minutes left to play.