A late Birkir Mar Saevarsson own goal denied Iceland a first-ever victory at a major tournament as Hungary stayed top of Group F with a 1-1 draw in Marseille.
Gylfi Sigurdsson had earlier scored from the penalty spot and a resolute defensive performance gave Iceland hope of a win that would have sent them to the top of the group.
But Saevarsson's late error, turning a cross into his own net, ensured that the spoils were shared.
The result means that Hungary need just a draw from their final group game against Portugal to seal a place in the knockout stages, while Iceland require a win against Austria to guarantee progression to the round of 16.
Iceland went into the game with an unchanged line-up from the team that drew 1-1 with Portugal in their first-ever game at a major tournament.
But it was Hungary who started the brighter of the two sides, having made three changes from the starting XI that beat Austria 2-0 in Bordeaux in their opening match of Group F, with Zoltan Stieber, Adam Lang and Tamar Priskin coming into the team.
However, the game turned on its head in the 39th minute when Gabor Kiraly spilled a routine take inside a crowded penalty area. The referee pointed to the spot after a number of blue-shirted Iceland players were felled, and while the initial reaction suggested that the penalty had been given for Kiraly on Ragnar Sigurdsson, it was a later foul by Tamas Kadar on Aron Gunnarsson that made up the official's mind.
Gylfi Sigurdsson stepped up to take the penalty and, in front of the massed ranks of whistling Hungary supporters, picked out the bottom corner as he sent Kiraly the wrong way.
The second half was a backs-to-the-wall defensive effort from Iceland, who sat deep and attempted to protect their one goal advantage.
But the smallest nation to ever appear at a major international tournament were unable to cling on for the win, with Saevarsson forced to lunge in on his own line and unintentionally turn the ball home.
A Gylfi Sigurdsson free kick crashed into the wall deep in stoppage time, with veteran forward Eidur Gudjohnsen -- who had come off the bench late in the second half -- seeing his follow-up effort deflected narrowly wide in the final action of the game.
Undefeated Iceland's wait for a win at Euro 2016 goes on, but that will need to come to an end against Austria on Wednesday if their tournament is not to come to an end at the first hurdle.
Hungary are highly likely to reach the next stage with their tally of four points from two games.