The Rangers (26-21-6) have lost three games in a row since going 8-2-1 from April 9-27. They have three games remaining.
"Just another level of disappointment, really," New York coach David Quinn said. "Just didn't get enough done this year. Had some great moments, some great stretches, but it just wasn't enough."
Capitals forward Alex Ovechkin played one shift of 39 seconds before leaving his return game after missing four with a lower-body injury.
Washington coach Peter Laviolette said Ovechkin didn't make his injury worse by playing.
"He felt really good in the morning and went into the warmups and it was not quite as good," Capitals coach Peter Laviolette said. "We were already shorthanded (17 skaters), so it's not like we were making a lineup decision. He got into the game and the first shift he just didn't feel comfortable, so at that point, again, we're not going to push or risk. ... If he felt anything that didn't feel right or normal, then he was going to come out."
Daniel Sprong scored his fifth goal in five games, Nic Dowd and Garnet Hathaway each had a goal and an assist, and Vitek Vanecek made 23 saves to help Washington win for the first time in three games.
The Capitals (33-14-5), who have qualified for the playoffs, are tied with the Pittsburgh Penguins for first place in the MassMutual East Division. Washington has one game in hand.
"I think it is a gutsy win coming down the stretch," Capitals defenseman John Carlson said. "These are important points important games for us as a team, and I think there are a lot of guys battling extra hard. It wasn't the easiest of games the way it turned out. I liked the way we stuck with it and overcame some adversity at different points in the game."
Mika Zibanejad scored two goals, and Igor Shesterkin made 34 saves for New York.
Rangers forward Artemi Panarin didn't return for the third period because of a lower-body injury. He was thrown to the ice during a post-whistle altercation near the Capitals net at 7:40 of the second period that resulted in Washington forward Tom Wilson getting a double-minor for roughing and a 10-minute misconduct.
Panarin and Rangers forward Pavel Buchnevich also received roughing minors, as did Capitals defenseman Brenden Dillon.
"He's fortunate he is OK," Quinn said of Panarin.
Dowd gave Washington a 1-0 lead at 14:12 of the first period, and Conor Sheary scored 1:32 later to make it 2-0.
Zibanejad scored at 18:00 to make it 2-1. He tied the game 2-2 at 2:30 of the second period with a power-play goal, and Kaapo Kakko scored on a 2-on-1 with Alexis Lafreniere at 4:35 to give New York a 3-2 lead.