Erik Cole scored twice for Carolina, including once on the power play.
Dubinsky struck 3:50 in when he took a centering pass from Jaromir Jagr in the slot and beat goalie Cam Ward with a backhand flip just inside the right post.
Avery scored on a low slap shot from the right wing that sneaked under Ward's pads with 44 seconds left in the first period. New York defenseman Dan Girardi intercepted a weak pass in the Carolina zone and got the puck to a wide-open Avery, who scored his third goal in five games.
The Rangers went up 3-0 early in the second when Drury outbattled Carolina defenseman Frank Kaberle for a puck in the slot and scored. Ward stopped an initial slap shot, but the puck landed at Kaberle's feet. Drury scooped it up and put it in.
Carolina got on the board at 7:39 of the second when Cole sent a slap shot from the left wing that deflected off the stick of Girardi and past goalie Henrik Lundqvist to make it 3-1.
Cole brought the Hurricanes within a goal early in the third when he converted on the back end of a high-sticking double minor to Marc Staal. Cole was the recipient of Staal's high stick and went to the dressing room for treatment, but he was back in time to score.
Shanahan restored New York's two-goal lead with 12:48 left when he tallied from the right wing.
Lundqvist stopped 26 shots. Ward made 24 saves.
