Both sides knew their chance to advance to the latter stages of the competition had already gone.
Defeats to Germany and Latvia earlier in the competition meant neither side had a point going into the game.
But that did not prevent either team from giving it their all in an entertaining encounter.
Unterluggauer blasted a 20-foot wrist shot over the shoulder of Slovakia netminder Jan Lasak for the only score of the third period to win the game.
Simon Wheeldon had opened the scoring for Austria less than three minutes in.
But Slovakia's Richard Pavlikovsky poked in a rebound on the power play six minutes later to make it 1-1.
Dieter Kalt put Austria back in front when he stuffed a shot home five minutes into the second period.
But Slovakia's Richard Linter scored off a faceoff to bring it to 2-2 before Unterluggauer's winner.