Szalai rewarded his side for a highly proficient display when he netted his first goal for club or country for 18 months after 62 minutes.
Austria, who had hit the post in the first minute, then lost Aleksandar Dragovic to a second bookable offence and Stieber wrapped up victory in the Group F clash from a late breakaway.
The result was unexpected with Austria boasting a technically excellent forward line. They needed just 30 seconds to serve warning of their ability as David Alaba seized on a loose ball and lashed a left-footed shot against the woodwork from distance.
They won't forget this moment in a hurry!#HUN #EURO2016 pic.twitter.com/r2GxDwVaiN Dragovic on Tamas Kadar. Their last EURO victory was a 3-1 extra-time defeat of Denmark on 20 June 1964.#EURO2016 #HUN pic.twitter.com/zTHfW2a0up