New York was outshot 10-8 in the second, but Nelson made it 2-0 when he redirected Josh Bailey's shot from the slot for a power-play goal at 19:49.
"Sorokin came up big for us and, all in all, we found a way to lock them down and play good team defense," Bailey said. "Tonight was just about getting the win."
Anthony Beauvillier drove to the net off Nelson's lead pass and scored on a backhand at 10:51 of the third for the 3-0 final.
The goal came 31 seconds after Kyle Palmieri had a goal overturned when video review determined the Islanders forward interfered with Vejmelka by pushing his glove, which dislodged the puck.
"I liked our response after the disallowed goal. We came right back out and scored," Trotz said.
Despite the result, Tourigny said he liked the intensity of his players.
"We won battles, we were pretty good, we had good scoring chances, but we couldn't finish," Tourigny said. "We paid for our mistakes. But if we play like that every night, I like our odds."
The Coyotes begin a six-game road trip Monday at the Florida Panthers, who have won their first five games.
"It's the start of the year and we're getting used to a new system, but I think in every game we're getting better and better," O'Brien said. "We're going to get it, and we're going to start winning hockey games."
NOTES: Zdeno Chara of the Islanders played his 1,613th game and moved past Ray Bourque into fourth on the all-time list among NHL defensemen. He is 38 games from tying Chris Chelios' record of 1,651. … Bailey tied Bobby Nystrom for 10th on the Islanders all-time list with his 513th point. … The Coyotes have allowed seven goals in 11 shorthanded situations. … Arizona has been outscored 25-8, including 12-2 in the second period.
