The result left Hungary with everything to play for in their final match against bottom-side Greece on Sunday.
Northern Ireland are through with 20 points, and either Romania (17) or Hungary (16) will take the remaining automatic place with the third place team in contention through the playoffs.
The Faroes stunned the home crowd taking the lead against the run of play in the 11th minute when Roaldur Jacobsen finished a quick counter attack, turning his marker Roland Juhasz and cleverly shooting past Gabor Kiraly from 13 meters.
Hungary piled on the pressure and Adam Bodi should have scored just before the break but his tame effort from 10 meters was well saved by Gunnar Nielsen.
Bode then had a shot from close range, but Nielsen made a fine save but Bode did not miss his second chance, the striker nodding in from five meters after Tamas Kadar's inch-perfect left wing cross in the 63rd minute.
The one-man show had only just started though as Bode grabbed his second eight minutes later as the powerful striker fired home a rocket after beating two defenders.
The evening which started so brightly ended on a low for the visitors who had captain Atle Gregersen sent off in the last minute for arguing with the Hungarian bench.
(Editing by Mike Collett)
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