He beat Pavelec at 37 seconds when he took the puck into the zone, cut down the middle and across the slot, and roofed a backhanded shot from the left circle into the far corner to make it 3-1.
"Just tried going across the net, but you can't really just pick those spots," Barzal said. "It was just kind of lucky, found its way in."
Barzal scored again off his own rebound on for a 4-1 lead.
"He just makes great plays every time," Beauvillier said.
Beauvillier, who gave the Islanders a 1-0 lead at 6:06 of the first period and chased Pavelec when his second goal made it 5-1, has four points (three goals, one assist) in two games since playing two games in Bridgeport of the American Hockey League in an effort to re-establish his game.
He also played one game with Bridgeport this week, when the Islanders (and Rangers) had their five-day break. He's with the Islanders on an emergency recall because of lower-body injuries to forwards Andrew Ladd and Josh Bailey.
"I just feel more comfortable," Beauvillier said. "I want the puck more. I know I can make plays and have success. It's all in my head, I guess."
Hayes scored a game-tying power-play goal at 14:25 of the first, but Prince scored his first of the season two minutes later. Zibanejad cut the deficit to 5-2 at 9:05 of the third, but Hickey and Clutterbuck scored in the final 3:54.
