Shea Theodore won it for Vegas at 2:09 of overtime when he deked T.J. Oshie and scored with a backhand in front.
"It's just been good for me this year," said Theodore, who has three overtime goals this season. "Hopefully we keep it rolling."
Evgenii Dadonov had a goal and an assist for the Golden Knights (42-31-5), who moved two points back of the Dallas Stars for the second wild card into the Stanley Cup Playoffs from the Western Conference.
"We know the reality of the situation right now," Vegas defenseman Alec Martinez said. "Our back is against the wall. These are must-win hockey games. Obviously, to get a win tonight and get those two points is huge, but really proud of the way the guys handled everything. We had some adversity throughout the game. It was a little bit back and forth, but it feels good.
"The job isn't done yet. We can enjoy this one for tonight, but we all know what the task at hand is."
Robin Lehner allowed one goal on 13 shots before being replaced at the start of the second period by Logan Thompson, who made 13 saves.
"As a goalie partner, you don't want to be in that situation," Thompson said. "Robin is a massive part of our team. He loves everyone, so you never want to see that."
Evgeny Kuznetsov had a goal and an assist, and Ilya Samsonov made 26 saves for the Capitals (43-23-11), who moved into a tie with the Pittsburgh Penguins for third place in the Metropolitan Division.
"I liked the structure of the whole game," Washington coach Peter Laviolette said. "[Vegas] is a good team. They're deep with offense, and I thought we did a pretty good job defensively."