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Alexandru Maxim had the first attempt on goal for the hosts in Bucharest, but his effort from range went over the bar.
The next Romanian to try his luck was Razvan Rat, whose shot from outside the box was saved by the keeper.
Romania continued to dominate and Dragos Grigore headed Maxim's corner just wide of the post.
With a minute of the first half left to play the hosts took the lead as Raul Rusescu slotted home after being picked out in the box by former West Ham United loanee Rat.
In the 82nd minute, Hungary found the net with their first shot on target, as Dzsudzsak fired in a precise free kick to ensure that his side went home with a point.