The triumph, which sealed their 3-1 victory on aggregate in front of an ecstatic crowd in the Groupama Arena, guaranteed Hungary's first appearance in any championship finals since the 1986 World Cup in Mexico.
"I have to congratulate my team, the association president Sandor Csanyi, and also Pal Dardai, who I took over from and he left this team for me," said Hungary's German coach Bernd Storck who took charge in July.
"The players' approach was fantastic during those two games, they fought well and deserved to qualify to France."
The Hungarians built on their first-leg win in Oslo when Tamas Priskin curled in a delightful 14th-minute opener and Markus Henriksen's own goal on 83 minutes made the tie safe.
Henriksen atoned with a late goal but it was scant consolation for the Norwegians as the home side celebrated qualifying for the European Championship finals for the first time since 1972.
The toast of Budapest was Hungary's 39-year-old Gabor Kiraly, who made a series of superb saves over the two legs.
Ferencvaros's home ground erupted almost like never before when Priskin curled in a fabulous strike from 16 metres after a quick counter-attack to put Hungary ahead.
Kiraly had to make a fine save from Haitam Aleesami's low shot in the 27th minute after a flowing passing move from the visitors and Norway came even closer when Even Hovland's header from a corner hit the post in the 45th minute.
Balazs Dzsudzsak's deflected shot struck the crossbar in the 58th minute after a rare counter-attack by the hosts, who dropped deeper to defend their overall two-goal advantage.
Kiraly made another excellent save in the 76th minute when he parried Marcus Pedersen's close-range shot before the tie was effectively sealed seven minutes from time as Dzsudzsak's corner was bundled under pressure into his own net by Henriksen.
The AZ Alkmaar man then pulled one back for Norway in the 87th minute but it was too little, too late.
"Congratulations to Hungary, they were better than us over the two legs," said Norway coach Per-Mathias Hogmo.
"I am deeply disappointed with our performance, we could not score in Oslo, that was a big setback. All in all, we came up short."
(Editing by Ian Chadband)
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