Brady Skjei unleashed a shot from the point before the puck hit Vesey and found it's way past Jets goaltender Connor Hellebuyck to give the Rangers the win in a closely contested hockey game.
Winnipeg had two potential goals called back via video replay in the game.
The first came a mere 24 seconds into the contest when Tyler Myers appeared to score after taking a feed from Kyle Connor before moving into the middle of the left faceoff circle before unleashing a slap shot that beat Henrik Lundqvist high blocker side but was disallowed after a coach's challenge for being offside.
That didn't matter as Nikolaj Ehlers converted a Mark Scheifele feed, blowing a slapshot past New York goaltender Henrik Lundqvist just 42 seconds later, beating him glove side to make it 1-0 Winnipeg.
The goal was his 21st of the season.
Mats Zuccarello tied the contest with just over a minute remaining in the first period when Nikolaj Ehlers turned the puck over at the blue line.
Zuccarello scooped up the puck before moving in and firing the disc through Connor Helleuyck's pads at 18:54.
The goal was his 9th of the season.
There was no scoring in the second period.
Michael Grabner also tallied for the Rangers, who won their second game in a row (27-24-5).
Henrik Lundqvist turned aside 37 shots for New York.
Connor Hellebuyck made 25 saves for Winnipeg.
Dustin Byfuglien appeared to have tied the game with 1:22 remaining in regulation but it was ruled no goal as Mark Scheifele deflected the disc past Lundqvist using a high stick.
Michael Grabner added a late empty netter to seal the win for the Rangers.
New York continues a four-game road trip Tuesday in Minnesota while Winnipeg hosts Washington.
SHOTS-NEW YORK 28 WINNIPEG 38