Iceland looked on course for their first win at a major tournament but an own goal from defender Birkir Mar Saevarsson denied the minnows two minutes from time.
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Both teams impressed during their first game of the group but chances were few and far between during a competitive opening quarter which saw just one clear-cut effort on goal, with Iceland's Jon Dadi Bodvarsson heading over the crossbar from eight yards out.
Hungary were dominating possession but they were struggling to find a way through a determined Icelandic backline, and the minnows should have taken the lead on the break when Gylfi Sigurdsson found Johann Berg Gudmundsson inside the penalty area, but his shot was rushed and straight at Gabor Kiraly.
That let-off resulted in Hungary upping the tempo in the final third and Laszlo Kleinheisler flashed a left-footed strike just past the far post, but six minutes before half time Iceland went in front from the penalty spot.
Aron Gunnarsson was brought down after Kiraly had spilled a corner and Sigurdsson made no mistake from 12 yards out, sending the veteran stopper the wrong way to give his nation a slender lead at the break.
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Hungary came out for the second half on the front foot but they continued to struggle to get shots on target with Kleinheisler's wayward effort the best of the strikes before the hour mark.
Balazs Dzsudzsak seemed to have their best hopes of an equaliser but he was being restricted to set pieces from outside of the penalty area, with two efforts being easily dealt with by Hannes Thor Halldorsson.
Iceland were content to sit on the edge of their own 18-yard box as they grew ever nearer a landmark triumph but with two minutes left, Hungary found an equaliser after Nemanja Nikolic's low cutback across goal was turned into his own net by Mar Saevarsson.
There was almost late drama when a shot from Iceland legend Eidur Gudjohnsen was deflected marginally wide of the post but his side had to settle for a point in Marseille.