The play started with Rangers goalie Henrik Lundqvist setting the puck up for Brandon Dubinsky behind his own net. Dubinsky carried the disc into the Carolina zone, but lost control as he avoided a defender en route to the crease. Zuccarello, playing in just his sixth NHL game, collected the puck behind the net, came around the far post and lifted a shot over Hurricanes netminder Cam Ward.
The 23-year-old Norwegian also set up a third-period goal by Matt Gilroy.
Dubinsky had two assists and Lundqvist made 31 saves for the Rangers, who halted their losing streak at two games.
Eric Staal tied the game with 6:14 left in regulation for the Hurricanes, while Ward stopped 33 shots.
Carolina has earned at least a point in five straight games (3-0-2).