The Rangers moved within two points of the New Jersey Devils, who lost 2-1 to the Boston Bruins on Saturday, for second place. They are three points behind the first-place Carolina Hurricanes, who lost 4-3 to the Buffalo Sabres.
New York and New Jersey each has two games remaining; Carolina has three.
Shesterkin has three shutouts this season and 11 in his NHL career.
"The way he plays and gives us a chance to win every night, it's nice to help him out -- although it was stressful the last few minutes," Rangers forward Mika Zibanejad said.
Niko Mikkola scored his first of the season, and Zibanejad had two assists for New York (47-21-12), which is 3-0-2 in its past five games and 12-2-3 in its past 17.
"It was a good hockey game. We played pretty well, and good, solid defensively," Rangers coach Gerard Gallant said. "It's our last game on the road until the [Stanley Cup] Playoffs, so it was good to play well on the road.
I thought tonight was just a good solid effort. It wasn't a high intense game, but it was exactly what we needed."