Substitute Panagiotis Kone won it for Greece four minutes from time with a thumping header, some consolation for a campaign which the 2004 champions finish at the bottom of Group F on just six points.
Hungary had already been assured of at least a playoff place before the game, and finish third in Group F behind Northern Ireland and Romania.
Greece's interim boss Kostas Tsanas had given youth a chance by starting PAOK Thessaloniki midfielder Dimitris Pelkas and AEK Athens FC’ forward Kostas Mantalos, as well as young PAOK full-backs Stelios and Kostas Stafylidis.
That paid dividends after just five minutes when Kitsiou’s low cross found Stafylidis at the back post for Greece's first home goal of the campaign.
Hungary were almost level in the 22nd minute but Balázs Dzsudzsák’s deflected shot was directed just wide by striker Dániel Böde’s outstretched right boot.
But after Pelkas blazed a shot over the bar for Greece, Hungary drew level in the 26th minute when Gergő Lovrencsics beat Stafylidis to head past Greek goalkeeper Orestis Karnezis.
Greece began the second half in the same attacking vein, but it was Hungary who went ahead against the run of play, Krisztián Németh prodding home from barely a metre out after Dzsudzsák’s corner was flicked on by Ákos Elek.
Greece’s response was instantaneous, Panagiotis Tachtsidis sprinting through onto Mantalos’ pass to score just two minutes later.
Németh again, and Greece's Kostas Mitroglou, exchanged strikes to make it 3-3 before Kone's late winner. Even if Hungary had won they won have finished third as Romania's 3-0 win over the Faroe Islands ensured they finished second behind Northern Ireland.
(Editing by Patrick Graham and Mike Collett)
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