Ivan Perisic, who scored when Croatia beat Israel 2-0 on Wednesday, set up the opening goal for Juventus striker Mario Mandzukic after a good one-touch build-up in the 29th minute against Hungary.
Dzsudzsak came close to an equaliser when his fierce low shot in the 36th minute went just wide.
The home team applied more pressure in the second half and the midfielder did not miss the next time he had a chance, drilling a superb effort into the top corner of the net from 22 metres in the 79th minute.
"We played offensively in the first half but in the second we dropped back a bit and Hungary had more possession," said Croatia coach Ante Cacic.
Hungary boss Bernd Storck thought his side deserved their draw.
"I expected a bit more from my team in the first half, more chances," he explained.
"In the second half we had opportunities, played better and that pleased me because we were playing against a quality team."
Euro 2016 will be held in France in June and July.
(Writing by Zoltan Fazekas; Editing by Tony Jimenez)
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