Kyle Palmieri blocked Zac Jones' shot from the point. The puck popped up in the air, hit Cuylle on the right side of his head and went into the net.
"I was just trying to get to the net, get in the goalie's eyes and I think it just got blocked," Cuylle said. "I saw it coming so I just had to get my face out of the way and whatever happens happens."
The Rangers went up 2-0 on Borgen's short-handed goal at 19:15. Cuylle took a hit from Noah Dobson high in the Rangers' defensive zone but still made the play to push the puck up the ice and out of the zone to Sam Carrick.
"I didn't think he was going to hit me because it was short-handed," Cuylle said. "I thought he was going to pull out. Obviously, nice read by Sam there."
Carrick carried the puck into the left face-off circle before leaving it back for Borgen, who scored with a bar-down wrist shot into the short side.
"You love to see him jump up in a play like that and he has the skill," Laviolette said. "That was a great shot and a big goal at the time."
Miller made it 3-0 at 3:43 of the third period with a blocker-side wrist shot from the right circle. The play was a result of Tony DeAngelo's neutral zone turnover. It was Miller's fifth goal in as many games.
"The third goal was not a pretty one, but we didn't generate," Islanders coach Patrick Roy said. "I thought both teams didn't generate much offense all night long. We didn't play the type of hockey that I was expecting."
Panarin scored into the empty net with 5:09 remaining in the third period to give the Rangers a 4-0 lead.
"I thought we were really good," Laviolette said. "'Shesty', he was good, he made big saves when he needed to, but I thought we were good in front of him. Guys were heavy in front of him. Blocked shots, heavy on sticks, eliminated second opportunities. When it got scrambly right in the slot area I thought our guys cracked down hard and defended well."
NOTES: The Islanders had a would-be goal from Lee at 8:24 of the first taken away because of a successful coach's challenge from the Rangers for offside. Video replay showed Casey Cizikas preceded the puck into the zone 10 seconds before Lee scored. … Brock Nelson played his 900th game. The center is the fifth to play at least 900 games with the Islanders, joining Bryan Trottier (1,123), Denis Potvin (1,060), Josh Bailey (1,057) and Bobby Nystrom (900). ... Miller's two points extended his point streak to six games (10 points; five goals, five assists). ... Rangers forward Chris Kreider missed his fifth straight game with an upper-body injury.