Beauvillier had played seven straight games without a goal and five without a point.
"It's just one of those nights where everything seems to go in, and you just kind of enjoy them and move forward after," Beauvillier said. "It's nice, especially when you've been struggling a little bit the past couple of games."
Josh Bailey had two goals and an assist for the Islanders (29-13-4), who scored three goals in the third period. Semyon Varlamov made 25 saves.
"The Rangers have been playing really well, and they keep finding a way to climb up the standings, so we wanted to do our best to push them back down," Bailey said. "All in all, good overall win tonight."
Kevin Rooney scored for the Rangers (23-17-6), who had won four in a row. Igor Shesterkin made 22 saves.
"They're a good squad," defenseman Brendan Smith said. "They're playoff ready, and we got a little taste of that for sure."
The Islanders tied the Washington Capitals for first place in the eight-team MassMutual East Division, one point ahead of the Pittsburgh Penguins. The Rangers are fifth, six points behind the fourth-place Boston Bruins, who won 2-0 at the Buffalo Sabres on Tuesday. The top four teams will qualify for the Stanley Cup Playoffs.
"We just weren't playing the way we needed to play if we were going to have a chance tonight," Rangers coach David Quinn said. "I thought we were sloppy, I thought we were too high-risk, too east-west. We weren't playing fast enough."
Bailey, who missed two games with an undisclosed injury, one-timed a feed in front from Jean-Gabriel Pageau to give the Islanders a 1-0 lead at 17:42 of the first period.