Clutterbuck made it 7-2 with his eighth of the season and fourth goal in the last four games with 1:05 remaining.
The Islanders won their second straight after a season-high six-game losing streak. They grabbed control with four consecutive goals after Hayes tied it at 1-1 at 14:25 of the first.
Prince scored his first goal of the season two minutes after Hayes’s power-play goal. The Isles then scored three more in the second.
Barzal made it 3-1 just 37 seconds into the period on a backhander past Pavelec and into the top right corner. He then scored his 15th of the season when he knocked in the rebound of his own shot at 4:11.
Barzal now leads all N.H.L. rookies in assists (29) and points (44).
Beauvillier got his second of the game and seventh of the season when he beat Pavelec from the left circle at 6:51. The Rangers then pulled Pavelec and sent in Lundqvist, drawing cheers from the crowd and chants of “Hen-rik! Hen-rik!”
The Islanders got on the scoreboard first. Eberle brought the puck up the right side and sent a pass across to Beauvillier, who skated forward and beat Pavelec from the left face-off circle at 6:06 of the first.
Hayes responded for the Rangers. Rick Nash, with two Islanders players on him, sent a pass from the left corner into the middle to Hayes, who put it past Halak from the inside edge of the right circle for his 10th.
Left wing Michael Dal Colle made his N.H.L. debut for the Islanders. … Islanders forward Casey Cizikas left with an upper-body injury in the second period. … The Islanders won, 4-3, in a shootout at Madison Square Garden on Oct. 19. The teams play two more times — both at Barclays Center — on Feb. 15 and April 5. … The Rangers scored a power-play goal for the third time in five games, but are just 3 for 30 over their last 10 games. … The Rangers next play at Pittsburgh on Sunday night, while the Islanders are at Montreal on Monday night.