Mats Zuccarello scored twice for the Rangers (5-3-0), who had a three-game winning streak end.
Cam Ward made 28 saves, and Victor Rask had two assists for the Hurricanes (2-3-2).
The Rangers made it 1-0 on Zuccarello's power-play goal at 8:31 of the first period. Mika Zibanejad banked a shot off Ward's back then cleared the rebound to Zuccarello, who hit the net from the right circle.
Skinner tied the game 1-1 with his third goal of the season at 12:34. His shot missed wide, but after Noah Hanifin settled the puck, Skinner gathered it and circled the net to beat Henrik Lundqvist off the blocker. Skinner was returning to the lineup after missing one game with a middle-body injury. It was his first game serving as an alternate captain, a designation he will have for all home games.
"He's got the ability to make things happen in tight spaces," Hurricanes coach Bill Peters said. "He's a goal-scorer. There's not a lot of them anymore. He's one of those guys that can hang in the weeds a little bit and all of a sudden he gets the puck at the right time and the right spot and it's in the net."
Skinner gave the Hurricanes a 2-1 lead 33 seconds into the second period when he took a stretch pass from Brett Pesce to score on a backhand from the slot. Skinner has four goals and nine points in six games.
"It's tough to pinpoint why, otherwise you'd try to stay in that for the whole year," Skinner said. "You go through stretches where you're in a rhythm, but I think our line's been clicking well all year. For me, it's about getting in positions where I can get off quality shots."