Vesey recorded his fourth career multi-goal game by scoring in the first and second periods. He added an assist on defenseman Neal Pionk's first career goal.
Chris Kreider and Mika Zibanejad also scored as the Rangers snapped a three-game losing streak while beating Buffalo for the 13th time in the last 16 meetings.
Rookie Alexandar Georgiev made 21 of his 43 saves in the first period and came within 4:34 of his first career shutout. Rangers goalie Henrik Lundqvist was not active for the game after taking a scary hit in a collision during the third period of Tuesday's loss to Columbus.
Robin Lehner gave up the first four goals on 12 shots before getting pulled. Linus Ullmark made 10 saves as Buffalo lost its fourth straight and dropped to 1-5-1 in its last seven.
Sam Reinhart scored a power-play goal for Buffalo, which has been outscored 16-2 during its losing streak.
Kreider made 1-0 with his 15th goal with 3:31 remaining in the first period, slotting in a rebound after Zibanejad's shot went wide of the net.
Vesey made it 2-0 with 1:58 remaining in the first after a Buffalo turnover behind the net. Mats Zuccarello fired a long pass that Lehner moved into position to play, but he also left an opening in the left side of the crease for Vesey's tap-in.
Pionk scored 3:59 into the second by lifting a slap shot from up close over Lehner after taking a long pass from Vesey. Zibanejad scored his 27th nearly five minutes later by tipping in Kreider's shot, extending his points streak to eight games.
Vesey made it 5-0 with a wrist shot late in the second during a power play.
--Field Level Media
Buffalo | NY Rangers | |
Sam Reinhart 1 | Points | Jimmy Vesey 3 |
Sam Reinhart 1 | Goals | Jimmy Vesey 2 |
Ryan O'Reilly 1 | Assists | Kevin Hayes 2 |
Sam Reinhart 1 | Power Play Goals | Jimmy Vesey 1 |
N/A | Short Handed Goals | N/A |
Linus Ullmark .909 | Save Percentage | Alexandar Georgiev .977 |
Linus Ullmark 10 | Saves | Alexandar Georgiev 43 |
Team | Shots | Goals | Power Play | Penalty Kill | Penalty Mins | Face Offs Won |
Buffalo | 44 | 1 | 1-3 | 1-3 | 11 | 32 |
NY Rangers | 23 | 5 | 2-3 | 2-3 | 11 | 25 |