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The hosts had an early chance inside the opening two minutes to take the lead, but Attila Fiola could not find the target with his effort.
Pukki came close again moments later, but he fired his strike from outside the box inches wide of the target.
Following a goalless first half, Zoltan Gera had an effort saved by the keeper minutes after the break.
Gera threatened again after connecting with Fiola's cross, but Lukas Hradecky saved his header.
With six minutes of the game left to play, Hungary made the breakthrough as Gera was picked out by Gergo Lovrencsics and headed an effort into the bottom corner.