A victory would have qualified Northern Ireland for its first major tournament since 1986, but a loss looked likely when Richard Guzmics volleyed in from close range in the 74th minute after goalkeeper Michael McGovern spilled a free kick.
Chris Baird was then sent off for picking up two yellow cards in the same phase of play, but Lafferty rescued a point by lashing home a rebound in the third minute of stoppage time for his seventh goal in qualifying.
"It was nerve-wracking, to say the least," Northern Ireland manager Michael O'Neill said. "It wasn't vintage, but it was pure theater."
Northern Ireland stayed top of Group F, a point ahead of Romania after its 0-0 draw against Greece, and is now sure of at least a place in the playoffs courtesy of a guaranteed top-three finish. Hungary is four points off the lead with two rounds remaining.
The top two teams qualify automatically. The best third-place finisher in the nine groups also qualifies, with the other teams in third place competing in the playoffs.
In the group's other match, Finland beat the Faeroe Islands 1-0 on Joel Pohjanpalo's 23rd-minute goal.
Northern Ireland can qualify by beating Greece at home in its next game, but will be without Lafferty because of suspension after the striker picked up a booking against Hungary. Baird will also be banned.
Northern Ireland was only among the fifth-seeded teams in the qualifying draw, but has lost just once in eight group matches.
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