Alexis Lafreniere broke a tie in the third period for the Rangers (26-18-6), who won their third straight to sweep the two-game set.
"When you play more and when you play a lot you grow more confident, you get more trust, you help the team win," said Shesterkin, who has started 16 of 19 games since returning from injury. "The guys block a lot of shots in front of you, and that's also helpful. Today we played a very smart hockey game defensively and offensively and the result was as it should have been. We're giving ourselves a chance to fight for the playoffs."
The fifth-place Rangers are four points behind the fourth-place Boston Bruins, who have two games in hand. The Bruins defeated the Pittsburgh Penguins 3-1 on Tuesday. The top four teams will qualify for the Stanley Cup Playoffs.
Sam Reinhart scored, and Ukko-Pekka Luukkonen made 38 saves for the Sabres (13-30-7), who lost 6-3 here Sunday.
"We did a good job creating chances. We definitely could've limited a few more for them," Buffalo center Casey Mittelstadt said. "But we created chances and the goalie played well. There's sometimes where we've got to bear down and finish some of those. I guess that's a positive. You always want to be creating chances. Some nights they're going to go in and some nights they're not."
Lafreniere scored on a one-timer from the high slot to give the Rangers a 2-1 lead at 9:49 of the third period.
Mika Zibanejad, at the left side of the net, extended it to 3-1 on the power play at 13:49 with his 20th goal of the season.
"I don't think we were too happy with our first [period]. I thought they came on really well," New York defenseman Brendan Smith said. "We knew what we had to do, we talked about it in the dressing room, and from the second it just continued to get better.
"It was kind of a trickle-down effect of just what we had to do and get better at and come through the system, and it took care of us. That's what we've been preaching. If you follow the system, it will take care of us. Ultimately, the third period became our best because we wore them down."