Florida (21-21-8) is nine points behind the Columbus Blue Jackets for the second wild card into the Stanley Cup Playoffs from the Eastern Conference.
Max Pacioretty scored and Fleury made 22 saves for the Golden Knights (29-21-4), who had three three-game losing streaks in their inaugural season and one earlier this season.
With Vegas' next game on Tuesday against the Tampa Bay Lightning, who have the best record in the NHL, coach Gerard Gallant said his message to his players was simple.
"Get ready for the next one," he said. "We can't do nothing about tonight or the last three games. We've got to get ready for the next game and play hard. We're going to play against the top team in the League in Tampa Bay, so it's going to be a battle, but hopefully that'll get us back to where we want to be."
Barkov gave Florida a 1-0 lead by beating Fleury with a wrist shot from the right circle 12 seconds into the game. It was the fourth-fastest goal from the start of a game in Panthers history and the fastest in a home game.
Pacioretty made it 1-1 at 8:07 with a shot between Reimer's legs after he deked past diving defenseman Mike Matheson.
Florida's Evgenii Dadonov was awarded a penalty shot with 58 seconds left in the first period after being hooked from behind by Brayden McNabb, but his shot went off the crossbar after hitting Fleury's thigh.
Despite being outshot 14-6, Florida had the best scoring chance of the second period when Jonathan Huberdeau had a breakaway, but Fleury stopped him with a pad save.
"We're allowing the big chance right now," Pacioretty said. "Obviously against teams with firepower like them, they're going to make you pay. We had opportunities to go up and then to tie as well and we just weren't able to do so."
