Myhrer, the defending slalom champion and Vancouver Olympics bronze medalist, won a World Cup slalom race for the fifth time in his career.
He led after the first run and did just enough to keep his lead, winning in 1 minute, 49.55 seconds to beat Hirscher by 0.06 seconds. Jens Byggmark of Sweden was 0.61 seconds behind in third.
American Ted Ligety, who won the opening giant slalom World Cup race in Soelden, Austria, by the biggest winning margin since 1979, tied for 13th place.
Ligety is now third in the overall standings. Hirscher leads with 140 points, with Italy’s Manfred Moelgg a close second.
The skiers were racing more than 100 miles north of the Arctic Circle, the world’s northernmost World Cup venue. Last year, the circuit races there were canceled because of lack of snow.
Myhrer said he was nervous going out last after seeing Hirscher put down a good time.
“This level is so high so if you take it easy just a little bit you’re behind,” he said. “It’s not easy to be alone up there ... knowing that everybody is down and it’s just you left. But I’ve proved that I can manage that.”
Hirscher trailed by 0.27 seconds after the first run and had the second fastest time of all skiers in the afternoon as well.
“I’m happy, really happy at the moment,” Hirscher said. “And six-hundredths of a second is really nothing.”
Byggmark had a top-three finish for the first time in four years World Cup racing. He burst onto the scene by winning back-to-back slaloms in Kitzbuehel, Austria, in 2007 but has not won a major race since. But he took silver at the world championships last year, and showed again Sunday he can still challenge.
“I’ve been working really hard to be more stable and I think I’ve reached my goal,” he said.
The Swedes have a new coach this season, former Alpine skier Johan Brolenius, whose training tactics have helped lift the team, Byggmark said.
“We’re fighting hard, and we’re helping each other,” he said.
The next men’s World Cup races are in downhill and super G in Lake Louise, Alberta, on Nov. 24-25.