Sergio Markarian was taking charge of Greece, European champions in 2004, for the first time and the 70-year-old Uruguayan made five changes from the team that suffered a humiliating 1-0 home loss to the Faroe Islands in November.
"We were close to winning the game, so maybe we lost two crucial points," former Panathinaikos manager Markarian said.
"We dominated most of the second half but could not score. All in all, I am satisfied with my team's effort."
Hungary captain Balazs Dzsudzsak stung the hands of Greece goalkeeper Orestis Karnezis with a low shot in the 20th minute, but there was little other goalmouth action in the opening 45 minutes.
Greece began the second half with more purpose and Stefanos Athanasiadis as well as Italy-based midfielders Panagiotios Kone and Yiannis Fetfatzidis went close.
PAOK striker Athanasiadis missed a golden opportunity in the 69th minute when he failed to connect with a cross while Hungary’s best chance was a long-range shot in the 88th minute from Adam Pinter.
Karnezis tipped the ball over the bar and substitute Nemanja Nikolic blazed over from the subsequent corner.
The result leaves Greece last in Group F with two points from five games and with only an outside chance of qualifying for Euro 2016. Hungary are third with eight points, behind leaders Romania on 13 and Northern Ireland on 12.
Hungary coach Pal Dardai said their opponents were better than their position in the group suggests.
"The first half was okay, we had two great chances. For most of the second half the Greeks dominated, but they are a far better team than their previous results suggest," he told a news conference.
Markarian took over as Greece coach last month after his predecessor Claudio Ranieri was sacked in November after getting off to a terrible start.
He took just one point from the national side’s first four Euro 2016 qualifying games, despite inheriting a team that made it to the last 16 of the World Cup in Brazil last year.
(Editing by Toby Davis)
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