Kaapo Kakko scored two goals and Igor Shesterkin made 31 saves for the Rangers (25-18-6), who have won two straight and are four points behind the Boston Bruins, who have two games in hand, for fourth place in the MassMutual East Division. The top four teams will qualify for the Stanley Cup Playoffs.
"That's the only thing we can do, try to win games and see how far that takes us, really," Zibanejad said. "That's our mindset right now, just focus on the next game, try to get the two points and move on from there."
The Rangers knew prior to the start of their game that they could close the gap in the standings with Boston losing 1-0 to the Pittsburgh Penguins earlier in the day.
"I thought you could feel it before the game that we understood that we had another opportunity to maybe climb back into this thing," New York coach David Quinn said. "There's hope."
Sam Reinhart had a goal and an assist, and Dustin Tokarski made 23 saves for the Sabres (13-29-7), who are last in the division and have already been eliminated from playoff contention.
Buffalo and New York will complete the two-game set here Tuesday.
"We did make some adjustments as the game went along, gave up a lot less, but obviously we couldn't make up the goal differential and that was the problem," Sabres coach Don Granato said. "We're obviously an evolving group here. I don't want to say learning, it's more evolving and adjusting. We made some adjustments in-game that slowed that tide and tipped things at least from a chance standpoint in our favor. But, again, too late and now we get a chance to play them right away again and it will be nice to see how we can respond."