Hungary was on the brink of defeat with 27-26 and Iceland was awarded a penalty in the closing seconds of regulation time. However, Snorri Steinn Gudjonsson missed the seven-meter shot, allowing Lekai to score on the counterattack.
"You know it's never over with someone like Nandor on goal," Hungary left back Barna Putics said, referring to goalkeeper Nandor Fazekas, who saved both penalties Iceland got.
After the first 10-minute overtime, the teams were level at 30-30, with Hungary finally edging ahead.
"It was a lottery in the end," Hungary coach Lajos Mocsai said. "We should have won in the normal 60 minutes instead of making such a Hitchcock drama out of it. We clearly were the stronger team."
Hungary raced into a 5-1 lead and stayed ahead of Iceland for almost the entire match. Hungary opened a five-goal gap to lead 17-12 early in the second half before the 2008 silver medallists , who won all matches in the group stage, rallied to lead 27-26.
"It's a very sad way to lose after an unbelievable match," Norway coach Gudmundur Gudmundsson said. "I am proud of the team and the way we played this tournament."
Hungary, which advanced from the group stage despite losing three matches, will play Sweden or Denmark for a place in Sunday's final.