Gylfi Sigurdsson’s nerveless penalty just before halftime had looked set to lift Iceland top of the Group F standings but Saevarsson turned in a cross by Hungarian substitute Nemanja Nikolic two minutes from the end of a match short on quality.
Hungary top the standings with four points, two ahead of Iceland and three clear of Portugal, who play bottom side Austria later on Saturday.
Hungarian fans and police forces briefly scuffled in the stands before kickoff.
Iceland, lifted by a gritty 1-1 draw against Portugal in their opening match, started well and Jon Dadi Bodvarsson headed the first chance of the match narrowly over the bar.
Johann Gudmundsson worked a shooting opportunity that was well saved by the outstretched leg of Gabor Kiraly but the 40-year-old Hungary goalkeeper quickly turned from hero to villain.
Kiraly failed to catch a routine cross and in the resulting goalmouth scramble defender Tamas Kadar was adjudged to have fouled Aron Gunnarsson.
Sigurdsson was coolness personified as he sent Kiraly the wrong way and tucked the ball into the corner of the net, prompting wild celebrations from the blue-clad Iceland supporters in the ground.
The second half followed the same pattern as the first.
Iceland forward Kollbein Sigthorsson headed a good opportunity just wide and Hungary midfielder Laszlo Kleinheisler lashed a couple of fierce shots well over the bar.
Hungary, impressive 2-0 winners over Austria in their first game, looked to have run out of ideas until Nikolic found space on the right and delivered a teasing cross which Saevarsson, under huge pressure, could only poke over his own line.
Iceland had an opportunity to win the game deep into stoppage time when they won a free kick.
Sigurdsson’s effort came back off the wall and substitute Eidur Gudjohnsen’s follow-up shot was deflected just wide of the post.