New York snapped a three-game losing streak with the win, while Carolina lost for the third time in four games.
Vesey scored the go-ahead goal with 7:45 left in the game, beating Scott Darling short-side for his 14th goal of the season. Vesey's goal came less than four minutes after a Victor Rask goal tied the score at three 8:23 into the period.
Zuccarello added an insurance goal for the Rangers five minutes later on an empty-netter off an assist from Ryan Spooner, and Vesey added his third on an empty-net goal with less than a minute left in the game.
Carolina's Lee Stempniak scored in the second period, sandwiched by two New York goals at the 1:52 and 11:51 marks, on goals by Vesey and Vladislav Namestnikov respectively. Pavel Buchnevich assisted on both Rangers goals, and Mika Zibanejad picked up his second assist on the Namestnikov goal.
The Rangers got on the board first at the 9:06 mark of the first period as Zuccarello banged home a rebound past Darling to give him goals in three straight games and 13 overall on the season.
Carolina tied the score at the 14:32 mark of the first period on a Teuvo Teravainen goal with assists from Brett Pesce and Sebastian Aho.
The Rangers won the game despite being outshot 44-23 and committing 19 giveaways.
The Hurricanes return home on Tuesday for two games, starting with a battle against Boston before facing Philadelphia as they fight for a playoff berth.
New York remains home on Wednesday against Pittsburgh before heading on the road Saturday to face St. Louis.
--Field Level Media
Carolina | NY Rangers | |
Victor Rask 1 | Points | Jimmy Vesey 3 |
Victor Rask 1 | Goals | Jimmy Vesey 3 |
Sebastian Aho 1 | Assists | Pavel Buchnevich 3 |
Victor Rask 1 | Power Play Goals | Vladislav Namestnikov 1 |
N/A | Short Handed Goals | N/A |
Scott Darling .810 | Save Percentage | Alexandar Georgiev .932 |
Scott Darling 17 | Saves | Alexandar Georgiev 41 |
Team | Shots | Goals | Power Play | Penalty Kill | Penalty Mins | Face Offs Won |
Carolina | 44 | 3 | 1-3 | 2-3 | 6 | 45 |
NY Rangers | 23 | 6 | 1-3 | 2-3 | 6 | 30 |