In addition, Schaefer's goal was his fourth game-winner this season, moving him past Denis Potvin for the most by a rookie defenseman in team history.
He also tied Vladimir Malakhov (1992-93) for the second-most goals in a season by an Islanders rookie defenseman behind Potvin (17 in 1973-74).
"Saying all those names, it's crazy because they're going to go down as some of the best defensemen to ever play," Schaefer said. "Obviously, Potvin playing for the Islanders is pretty cool. Everyone knows him. He's created a great legacy with the Islanders and everything. You don't really think about it that much, but when those names pop up and you hear those names, it's an honor to be with those guys and things like that and see what they've done."
Carson Soucy scored against his former team and Ilya Sorokin made 20 saves for the Islanders (30-19-5), who also defeated the Rangers 5-2 at UBS Arena on Wednesday, 5-0 at Madison Square Garden on Nov. 8 and 2-0 at home on Dec. 27.
It's the first time since 2017-18 that the Islanders have swept the regular-season series against the Rangers.
"I think from just the start when I played my first Islanders-Rangers game, the building is full of both fans and it's loud," Schaefer said. "We can score here, and it gets loud because we've got a lot of fans here. They can score there (UBS Arena), and it can get a little loud. But our fans are way louder than theirs. It's pretty fun when we go on the board and our fans are cheering."
The Islanders have won three games in a row and are 5-2-1 in the second game of a back-to-back this season.
"It's a good win for us," Islanders coach Patrick Roy said. "It was not our best game of the season but that's what good teams do, they find ways to win. I thought we played well enough defensively to allow ourselves to stay in that game and score those two late goals in the second period. And Ilya was good again."
Mika Zibanejad scored and Jonathan Quick made 19 saves for the Rangers (22-27-6), who have lost five of their past six games and 10 of 12.
New York played its second game without Artemi Panarin, who is being scratched for roster management purposes because he is expected to be traded.
"I thought for the most part the team competed pretty hard," Rangers coach Mike Sullivan said. "I thought we generated a fair amount of scoring chances. The goals are hard to come by now. I thought Mika's line in particular had a number of Grade A looks. Give Sorokin credit, I thought he made some timely saves for them. There's a fine line between winning and losing; I thought we played well enough to win."
Soucy scored at 17:18 of the second period to give the Islanders a 1-0 lead.
Maxim Shabanov got the puck across the zone to Soucy in the left circle. He snapped a shot that snuck through Quick and into the net on the short side.
"Kind of a lucky one, but ultimately just nice to contribute," Soucy said.