Reilly Smith and Alex Tuch scored in the third period for Vegas (35-13-4). Tuch's goal broke a 3-3 tie at 14:48.
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"It's big for our confidence and just a learning experience for our team too, knowing that we can win these games," Fleury said. "We're down by one, we're down twice, three times and we always played our way back. I was really confident about this game that our guys would play great from start to finish."
Nicklas Backstrom, Chandler Stephenson and Matt Niskanen scored for Washington (30-17-5), which has lost its past two games. Philipp Grubauer made 27 saves.
"Halfway through the third [period], we got a little passive," Backstrom said. "Let them back in the game and they won it. A tough one."
The Golden Knights have won five of seven games and are 20-4-3 since Dec. 3. They rebounded from a 5-2 loss to the Minnesota Wild on Friday and are 3-1-0 thus far on their six-game road trip.
"It was a good bounce back game for us after our last game, a real tough loss," Golden Knights coach Gerard Gallant said. "We didn't play that well. But today, everybody competed hard and everybody played the game that we're supposed to play. It's a huge win for us."
Capitals coach Barry Trotz challenged Tuch's goal for goalie interference, but the call stood after video review.
"Every time you turn the puck over, they come right back down your throat," Grubauer said.
Backstrom gave the Capitals a 3-2 lead 52 seconds into the third period.