Hungary ended 30 years of hurt by defeating Norway 2-1 in Budapest on Sunday to seal a 3-1 aggregate victory and a place in Euro 2016.
With a one-goal lead in hand from the first leg, Tamas Priskin put the home side in prime position to progress when he found the top corner early on with a fine finish, while Even Hovland headed against the post for Norway.
Despite the visitors needing two goals after the break to qualify, it was Hungary who looked more likely to find a second and it arrived late on through an own goal from Markus Henriksen.
Henriksen struck a late consolation at the other end, but visiting coach Per Mathias Hogmo was left to reflect on a toothless performance in such a pivotal game.
The famous triumph sends Hungary through to their first major tournament in 30 years – their last appearance coming at World Cup 1986 – and their first qualification for the European championships since 1972.
Norway, meanwhile, will have to wait until qualifying for the World Cup in 2018 for another chance to end their wait – which has gone on since Euro 2000 – for an appearance at a major international event.
After he was an unused substitute in the first leg, Real Madrid youngster Martin Odegaard was brought into Norway's starting line-up and Vegard Forren was fit to play after a throat infection. The hosts were missing Zoltan Gera through suspension.
Hungary made an excellent start, taking a 14th-minute lead. A long ball down the left from defender Tamas Kadar was gathered by Priskin and the striker cut inside to fire a magnificent right-footed finish into the top corner with Forren and Hovland failing to get a challenge in.
Norway threatened for the first time midway through the opening half when Haitam Aleesami was denied by Gabor Kiraly racing off his goal-line to make a save following a slick passing move, while Hovland headed the resulting corner wide.
The home side knew that a second goal on the night would all but end the tie as a contest and Gergo Lovrencsics should have done better when he headed off target after Laszlo Kleinheisler whipped in a threatening cross.
Norway thought they had got back into the game on the stroke of half-time, but Hovland's header from Odegaard's corner struck the foot of the post, meaning the visitors went into the break knowing they still needed to score at least twice to rescue their fading Euro 2016 hopes.
The visitors made two changes for the second half, with Odegaard and Mohammed Elyounoussi replacing Marcus Pedersen and Pal Helland.
It was almost game over when Hungary captain Balazs Dzsudzsak led a counter-attack and saw his left-footed shot take a deflection before crashing off the top of the crossbar.
Dzsudzsak was desperate to seal the result and a driving run led to him shooting with his other foot, forcing Orjan Nyland to get down brilliantly and push it wide.
Despite the desperate nature of their situation, Norway struggled to make an impact going forward until Kiraly saved low from Pedersen in the closing stages after Alexander Tettey shot had fallen into his path.
Any hopes of a comeback were ended when Hungary scored a second in fortuitous fashion. Henriksen diverted a powerful corner towards his own net and it flew in via a combination of the post and Nyland.
Norway did finally net when substitute Jo Inge Berget's shot was parried by Kiraly and Henriksen followed up to finish at the right end, but it was too late for a comeback as Hungary fans and players alike went wild at the full-time whistle.