With the teams tied, 3-3, in the third period, Vesey tracked down a pass from Buchnevich while breaking down the left wing into Carolina's zone, and fired a wrist shot that beat Scott Darling at 12:15 of the final period that finally put the Rangers up for good.
"That's where I was going," said Vesey of his shot. "I had a shot from a similar angle in the first period and kind of misfired and put it lower than I wanted. I was definitely trying to go high blocker and just fortunately it went in."
Vesey completed the hat trick with an empty-netter with 24 seconds remaining in the game.
New York opened the scoring 9:06 into the contest when Mats Zuccarello scored his third goal in as many games. Neal Pionk's point blast missed the net, but Zuccarello scooped up the fortuitous bounce on the side of the net, which the winger potted for his 13th goal of the season.
Teuvo Teravainen tied the game before the end of the period, but Vesey again put the host in front just 1:52 into the middle frame. Buchnevich won the faceoff by pushing the puck forward and sending a backhand pass to Vesey, who was stationed on the side of the net for the easy tally.
Carolina, though, evened things again when Lee Stempniak beat Alexandar Georgiev at 7:48.
Namestnikov put the Rangers ahead for the third time with a power play goal at 11:51 of the second. Buchnevich's shot from the slot caught a piece of Namestnikov and found a lane through goaltender Scott Darling.
But Carolina tied it for a third time on the power play when Victor Rask's slap shot from the point beat Georgiev at 8:23 of the third period.
After Vesey's goal put the Rangers ahead for the fourth time, Zuccarello netted his second of the night into the empty net with 2:37 remaining, before Vesey's third into the open cage.
Georgiev picked up his second win in as many starts, and finished the game with 41 saves to become the first Rangers rookie goaltender since Dan Blackburn to record at least 40 saves.
Georgiev, who now has five games under his belt, said he's growing more comfortable - and confident - with each passing day as a Ranger.
"I try to keep the same emotions and my preparation," Georgiev said. "I would say I feel a bit more confident, but pretty much the same guy, same goalie just playing my game."
Coach Alain Vigneault said simply it was the Georgiev's strong performance, coupled with a difficult night for Darling, that proved to be the difference.
"He's played well for us," Vigneault said of the 22-year-old netminder. "He's done what you want your goaltender to do - give yourself a chance to win, give your team a chance to win and that's what Alex is doing. And that's what we need from our goaltending, because on a lot of nights that's the difference in this league."
