Whitecloud muscled a long-range shot past Panthers goalie Sergei Bobrovsky at 6:59 of the third period to put Vegas ahead 3-2, his shot through traffic finding its way under Bobrovsky's glove.
Captain Mark Stone added an insurance goal at 13:41, the goal standing up after a review for a high stick on Stone, who had knocked the puck out of the air.
"Obviously the first one's huge," Whitecloud said. "But you take a little bit to enjoy it, then move on to tomorrow."
Former Panthers Jonathan Marchessault and Reilly Smith both scored for the Knights and defenseman Shea Theodore had a goal and an assist.
Vegas goalie Adin Hill made 33 saves -- including a spectacular full-stretch stick save of a shot from Nick Cousins early in the second period that kept the score level at 1-1.
"I guess that's kind of the stuff you dream of when you're growing up, making that big save or if you're a forward making that big shot in overtime to get the game-winner," Hill said.
"It was nice - just kind of desperation there, reached out my stick and got a piece of it."
Whitecloud said he couldn't tell at first from the bench if Hill had made the save, but he wasn't surprised.
"That incident right there is what Adin Hill is," Whitecloud said.
Center Eric Staal and right wing Anthony Duclair scored for the Panthers and Sergei Bobrovsky made 29 saves, but Florida saw their streak of eight road wins end.
They had also been 8-0 when scoring the first goal this post-season, but this time the Panthers couldn't make their early advantage hold up and the Knights will have a chance to double their advantage when they host game two of the best-of-seven series on Monday.
Each team is seeking a first Stanley Cup crown, and both are in the final for the second time.
The Panthers were swept by Colorado in 1996 and the Golden Knights fell to Washington in their 2018 inaugural campaign.
Florida, back on the ice for the first time in 10 days after their sweep of the Carolina Hurricanes in the Eastern Conference final, saw their streak of eight road victories end.
Staal gave the Panthers a 1-0 lead at 9:40 of the first period with a short-handed wraparound goal.
Marchessault made it 1-1 at 17:18 with a power-play goal and Theodore put the Knights up 2-1 at 10:54 of the second with a wrist shot - his first goal of the playoffs.
Duclair knotted it at 2-2 with 11 seconds left in the second period with a shot off Aleksander Barkov's face-off win.
Smith capped the scoring with an empty netter with less than two minutes to play.
