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Alex Ovechkin, Evgeny Kuznetsov and Devante Smith-Pelly scored, and Braden Holtby made 21 saves for the Capitals, who lead the best-of-7 series 2-1. Kuznetsov, who was injured in Game 2, had an assist, and Jay Beagle had two assists.
Game 4 is here Monday.
Washington's previous appearance in the Cup Final was in 1998, when it was swept by the Detroit Red Wings, losing Games 3 and 4 at home.
"The city is excited, fans are excited, but it's only two [wins]," Ovechkin said. "We just have to move forward and don't think about it too much."
Tomas Nosek scored, and Marc-Andre Fleury made 23 saves for the Golden Knights, who have lost back-to-back games for the first time in the Stanley Cup Playoffs.
The Golden Knights trailed a series once before, when they lost Game 1 to the Winnipeg Jets in the Western Conference Final. Vegas won the next four games.
"It doesn't mean we're out of it," Fleury said. "We still have a lot of minutes to be played, a lot of periods. Got a lot of time to put ourselves back in the series."
Video: VGK@WSH, Gm3: Smith-Pelly finishes Beagle's feed
The Capitals didn't give the Golden Knights much time or space early in the game. Even though they didn't score, the first period was arguably Washington's best period defensively in the series.
The Capitals blocked 15 shots, including four by defenseman John Carlson and two from forward Ovechkin, and won 13 of 18 face-offs. Ovechkin had three shots on goal, eight shot attempts and two hits.
"He set the tone with intensity, with physicality," Capitals coach Barry Trotz said of Ovechkin, the Washington captain.
The Golden Knights blamed themselves for being careless with the puck.
"You can live with the losses if you're just not getting the bounces, but the way we played isn't good enough for this time of year," Vegas forward Ryan Reaves said.
Ovechkin scored 1:10 into the second period to give Capitals a 1-0 lead. It was his 14th goal of the playoffs, tying him with John Druce for the Capitals record in a single postseason (Druce scored 14 in the 1990 playoffs).
Fleury stopped Carlson on a deflection, Kuznetsov from the high slot, and Carlson again from the low slot. The rebound came to Ovechkin, who scored on his backhand while falling over Vegas defenseman Brayden McNabb at the bottom of the left circle.
"I thought it was sort of the right thing in a playoff game, our first victory in the Final at home, that Alex would score the first goal," Trotz said. "I thought that was a little bit of poetic justice, if you will, for all the tough times."
Kuznetsov scored his 12th goal of the playoffs to give Washington a 2-0 lead at 12:50 of the second. T.J. Oshie blocked Shea Theodore's shot from the left point, and Beagle got the puck and moved it up to Kuznetsov, creating a 3-on-1 with Dmitry Orlov. Kuznetsov beat Fleury with a low blocker-side shot from the right circle.
Ovechkin, who was on the bench, was seen letting out a huge scream.
"It's just automatic," he said. "You just get excited."
Vegas got within 2-1 on Nosek's goal off a Holtby giveaway from behind the net at 3:29 of the third period, but Smith-Pelly gave Washington a 3-1 lead at 13:53.
Beagle took the puck from Theodore on the forecheck and passed to Smith-Pelly cutting to the net. He beat Fleury with a glove-side shot from the slot.
"I think we're getting stronger as this series goes on," Trotz said.
Video: VGK@WSH, Gm3: Oshie on Caps' effort, crowd in win
Goal of the game
Ovechkin's goal at 1:10 of the second period.
Video: VGK@WSH, Gm3: Ovechkin opens scoring to tie record
Save of the game
Holtby's save on Jonathan Marchessault at 5:50 of the second period.
Video: VGK@WSH, Gm3: Holtby denies Marchessault after deke
Highlight of the game
Kuznetsov's goal and Ovechkin's celebration at 12:50 of the second period.
Video: VGK@WSH, Gm3: Kuznetsov snipes wrister past Fleury
They said it
"We need to have some poise with the puck. We've got to make the right plays. We've got to make it easier on each other. Once we made a play, we're rushing the next one. We've got to have some poise and settle down a little bit. We've got to get back to what's made us successful." -- Golden Knights forward James Neal
"It's obviously fun for us to see the city like this and bring that kind of energy and enjoyment. The fans have been phenomenal. But in saying that, our group as professionals, you get more out of making that sacrifice so everyone can enjoy it. So we're just putting our head down, focusing on Game 4 to keep working toward that big goal." -- Capitals goalie Braden Holtby
Need to know
When the Stanley Cup Final has been tied 1-1, the winner of Game 3 has won the series 77.8 percent of the time (21 of 27) since 1939, when it went to a best-of-7 format. ... Game 3 was the 100th game in Vegas history (64-29-7). ... Ovechkin became the 34th player to score 60 NHL playoff goals. ... Washington finished with 26 blocked shots and won 39 of 62 face-offs (62.9 percent).
Game 4 of Stanley Cup Final at Capital One Arena on Monday (8 p.m. ET; NBC, CBC, SN, TVAS)
Video: Capitals win Game 3 to go up 2-1 in Stanley Cup Final
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