Jeff Skinner scored, and Cam Ward made 18 saves for the Hurricanes (35-33-11), who will miss the playoffs for the ninth straight season, the longest streak in the NHL.
"It's been a grind for quite a while here," Carolina coach Bill Peters said. "There was more expectation, more realistic expectation and a higher expectation."
Ryan Sproul and Kevin Hayes scored, and Henrik Lundqvist made 40 saves for the Rangers (34-36-9), who swept the four-game season series against the Hurricanes.
WATCH: [All Rangers vs. Hurricanes highlights]
Lundqvist, who is 19-3-0 in his past 22 games against Carolina, tied Harry Lumley for 14th place on the NHL list for games played by a goaltender (803).
"Henrik played awesome," New York defenseman Brady Skjei said. "There were a few shots where we could have been down early, but he kept us in the game. As defensemen, I thought we did a good job with our sticks. There were a lot of 1-on-1 battles, and I thought we won quite a bit of them."
Sproul scored to give the Rangers a 1-0 lead at 3:01 of the second period with a wrist shot in the slot off a pass from Paul Carey. The goal was Sproul's first since Nov. 15, 2016 when he played for the Detroit Red Wings.
"[Vladislav Namestnikov] on the wall made a great play to Carey, who made a great play to me," Sproul said. "When you're walking down the slot like that, it makes it pretty easy."
Hayes extended the lead to 2-0 with a shorthanded goal at 13:23. He scored on a one-timer off a pass on a 2-on-1 from Jesper Fast.