Brock Nelson and Anthony Beauvillier scored, and Ryan Pulock had two assists for New York (17-12-4), which has won three straight, including a 4-1 victory against the Colorado Avalanche on Monday.
"It's big on the road like that, against a couple of good teams. To get those two wins is important for us," Pulock said. "We knew we had to buckle down and be better, and I think the last two games we've done a better job."
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Lehner, who was 0-4-2 in his previous seven games, has allowed one goal in three of his past four starts.
"We finally got him some goals, which are a helpful way to win at hockey games," Islanders coach Barry Trotz said. "I thought he made some really key saves. I was really happy with him."
Mario Kempe scored his first goal of the season for Arizona (14-17-2), which is 1-6-0 in its past seven games. Darcy Kuemper made 29 saves and is 0-4-1 in his past five starts.
Kempe made it 1-0 at 9:14 of the first period, receiving a stretch pass from Alex Goligoski and scoring from the right circle. It was his first goal since Oct. 24, 2017 (23 games), also against the Islanders.
"They came out really well. I think they were the better team the first 10 minutes and they had some really good opportunities," Lehner said. "After 10 minutes we started playing better."