Stone extended the lead to 4-2 with an unassisted goal at 13:41. He knocked down Matthew Tkachuk's clearing attempt and scored from the slot to the glove side.
Florida challenged for a missed stoppage in play for high-sticking on Stone, but the goal was upheld after official review. Maurice said he challenged because it was close enough to being a high stick, the goal made it 4-2 and there was only 6:19 left.
Reilly Smith scored an empty-net power-play goal from the red line at 18:15 for the 5-2 final. Hill had denied Sam Reinhart on a breakaway 22 seconds earlier.
"It's a long series, lots of hockey ahead of us," Bobrovsky said. "We play, we learn and we move on."
The Panthers couldn't build on the momentum they took into intermission after Duclair tied it 2-2 with 10.2 seconds left in the second period on a shot off Aleksander Barkov's face-off win in the right circle.
"Yeah, we gave up a late goal, but I'd rather it then than at the start of the third to be honest," Cassidy said. "We have time to regroup and sort of say, 'Settle down and get back to work.' It was a good shot. I think it hit Whitecloud on the way in. But they generated some good looks, so it wasn't like we played this perfect period and it was demoralizing. For us, we do have a veteran group and I think it showed going into the third, and I think it showed late in the game when we were able to keep our discipline and get to the finish line."
Staal gave the Panthers a 1-0 lead at 9:40 of the first period with a short-handed wraparound goal.
Marchessault tied it 1-1 at 17:18 with a power-play goal on a one-timer from the slot.
Hill then came up with his first big moment of the Cup Final to deny Nick Cousins from in close on what looked like a sure goal 50 seconds into the second to preserve the tie.
"That was definitely a momentum swinger for sure," Stone said. "That got the bench energized. It kind of opens up your eyes a little bit too. Maybe we weren't playing great to start the second, but that save kind of turned it around for us and got us going in that period."
Theodore gave Vegas a 2-1 lead at 10:54 of the second with first goal of the playoffs. He scored on a wrist shot from above the hash marks through a screen from Brett Howden after he dangled and spun to get free from Duclair at the right point before taking the puck to the middle.
It was his first goal in 17 playoff games this season and his first in 28 games dating to March 7.
"I was looking for one," Theodore said, "and I guess tonight is a pretty good night to get the first one."
NOTES: Staal's goal marked the third time in Stanley Cup Final history the first goal of the series was scored short-handed. It also happened in 1997 (Kirk Maltby, Detroit Red Wings) and 1936 (Bucko McDonald, Detroit). … The goal was also Staal's first in the Cup Final in nearly 17 years (June 14, 2006, for the Carolina Hurricanes), the longest span between goals in Final history. … Marchessault extended his point streak to six games (eight points; five goals, three assists).