NEW YORK — With each passing game, it appears the New York Rangers may have found an apprentice goalie for next season.

Jimmy Vesey had two goals and an assist to support Alexandar Georgiev’s season-high 43-save performance and the Rangers beat the Buffalo Sabres 5-1 Saturday night.

Chris Kreider, Mika Zibanejad and Neal Pionk each added a goal and an assist as the Rangers completed a sweep of the three-game season series and improved to 13-3-0 in the last 16 meetings with the Sabres.

“The main thing is that we won the game,” said the 22-year-old Georgiev, who has been impressive since his recall from AHL Hartford last month after backup Ondrej Pavelec was injured. “We scored great goals in the first period, played really good defense. The guys were really battling to not let them have many good opportunities to score.”

Sam Reinhart spoiled Georgiev’s shutout bid with 4 1/2 minutes remaining for the Sabres, who have lost four straight. Robin Lehner gave up four goals on 12 shots before being replaced near the midpoint of the second period. Linus Ullmark came on and stopped 10 of the 11 shots he faced.

Kreider and Vesey each scored in the first before Pionk got his first NHL goal early in the second.

“It’s hard to believe it at first. It’s just such a great feeling when those guys come get you and they all give you a huge hug,” Pionk said. “Something I will never forget.”

Zibanejad made it 4-0 with his team-leading 27th at 8:45 of the second, giving the Rangers goals on four consecutive shots for the first time since Jan. 3, 2015, also against the Sabres, according to the Elias Sports Bureau.

Georgiev, starting for fifth time in six games, made 21 saves in the first period, nine in the second and 13 more in the third. He has four wins overall and could emerge as Henrik Lundqvist’s backup next season.

“I’m just trying to play my game and help the guys in every way I can,” he said.

The Sabres outshot the Rangers 21-10 in the first period but it was Kreider opening the scoring with 3:31 left with his 15th of the season with assists to linemates Zibanejad and Jesper Fast. The line has totaled 25 points in last five games.

Zibanejad has 12 points in his last seven games including eight goals. Kreider has a five-game points streak with three goals and seven assists. Fast has an eight-game point streak with three goals and seven assists over the span.

Vesey made it 2-0 with just under two minutes remaining. Mats Zuccarello had an assist for his team-leading 50th point.

Pionk scored his milestone goal at 3:59 of the middle period, extending his points streak to seven games. He has a goal and nine assists during the stretch. The 22-year-old defenseman was recalled from Hartford of the AHL last month and has 14 points overall.

Vesey’s 17th made it 5-0 with 5:13 left in the second.

“We lost coverages, guys were beating us up the ice, and I think it goes back to have that respect in our own zone and for our goaltender,” Sabres coach Phil Housley said “We have to keep the puck out of our net.”

The Rangers are 6-4-3 since dealing Rick Nash to Boston, and captain Ryan McDonagh and J.T. Miller to Tampa Bay as part of a promised purge of top players at the trade deadline. Though they will miss the playoffs for the first time since 2010, they have played with renewed energy since the deals were made.

“We had some grade-A chances that we were able to capitalize on and we got a lead,” Rangers coach Alain Vigneault said. “For the most part, we had some good looks and our goaltender was able to make some big saves.”

NOTES: The Rangers also defeated Buffalo 3-2 in overtime in the Winter Classic at Citi Field — which was considered a home game for the Sabres — on New Year’s Day, and won 4-3 at Madison Square Garden on Jan. 18. … The Rangers again scratched Lundqvist after he was shaken up in a collision last Tuesday against Columbus. … The Rangers also scratched F David Desharnais. … Buffalo scratched F Jacob Josefson, F Justin Bailey, D Justin Falk and D Nathan Beaulieu.

UP NEXT

Rangers: Host Washington on Monday.

Sabres: Visit Toronto on Monday.