Kleinheisler scored the game's only goal in the 26th minute, taking a touch to control Adam Szalai's pass before whipping the ball home after Norway had created a flurry of chances in a strong opening period.
A draw in the home leg on Sunday will put Hungary into the finals in Francen after years in the wilderness since they qualified for the 1986 World Cup.
"We are happy now after 90 minutes, to win this game, but the second game will be on Sunday. We can't think about celebrating," Hungary coach Bernd Storck told reporters.
"It's a very good job and nothing more -- we have to do a very good job on Sunday," he added.
Norway attacked relentlessly in search of an equaliser but they lacked guile in attack as Hungary's veteran goalkeeper Gabor Kiraly celebrated his 100th cap with a clean sheet.
Norway were denied two second-half penalties for fouls on Omar Elabdellaoui, and coach Per-Mathias Hogmo expressed his dissatisfaction with English referee Mark Clattenburg.
Clattenburg booked the full back for diving for the second incident and turned down an even stronger call for the first when Elabdellaoui went down after having his shirt pulled.
"Clearly a penalty. Crystal clear," Hogmo told reporters. "It's remarkable that an experienced referee didn't see it."
Norway battled on and came close late on, but substitute Pal Andre Helland's header hit the bar and Hogmo now has to rally his players for the second leg in Budapest.
The defeat was Norway's first by Hungary since 1981 and highlighted their inability to score, having managed only 13 goals during a qualifying campaign in which midfielder Alexander Tettey was their top scorer with three goals.
In all they managed 16 shots but only six on target, the best coming in the third minute when captain Per Ciljan Skjlebred's well-struck effort was steered to safety by Kiraly.
"It's about putting the ball in the back of the net, and we didn't manage that today," winger Jo Inge Berget told Reuters.
"It's a bit frustrating, as we could have easily got one today, but it didn't happen."
Hungary, on the other hand, did exactly what they set out to do, ceding possession and then striking on the counter-attack.
At the final whistle the Hungary players immediately went to celebrate with the small but vocal pocket of away supporters, some of them bare-chested despite the freezing temperatures.
(Reporting by Philip O'Connor; Editing by Ken Ferris)
Share
