Brandon Montour and Aleksander Barkov each had a goal and an assist after Vegas took a 3-0 lead. Sergei Bobrovsky made 28 saves for the Panthers.
"You've got to win four," Barkov said. "It's 3-1. I know they're one win away, we're three wins away, but all we can do is think about one game, bringing it back to Florida and that's our goal. That's what we want to do in Vegas, go there and try to win that game."
The Panthers pulled Bobrovsky with 2:26 remaining and got a power play with 17.4 seconds to play when Alex Pietrangelo was sent off for delay of game as he tried to send the puck around the end boards but it instead went directly into the netting.
Sam Reinhart's shot from the left face-off circle with nine seconds left broke Zach Whitecloud's stick, and Hill stopped Carter Verhaeghe's shot from the right circle with four seconds remaining. The clock expired before Matthew Tkachuk took the next shot from the slot.
"I like our fight," Montour said. "We're fighting back. In that third period we're a desperate team and we're dangerous. Obviously, they may be sitting back a little bit 3-0 but we've just got to keep pushing. We've got it in this group."
Tkachuk didn't play for a stretch of 10:41 in the third period, but he was on the ice for the final 2:26 after Bobrovsky was pulled.
He wouldn't discuss the reason he sat, but Florida coach Paul Maurice indicated it was an injury situation, saying Tkachuk was grinding and that the Panthers "were looking, hoping to get into a situation where he could use what he had to give us."
"Just trying to find a way to find a way out there to make it work tonight and came up just probably a second short," Tkachuk said. "Time ran out there with me and [Sam Bennett] whacking away. Two more seconds there, you never know."
Florida was also down 3-1 to the Boston Bruins in the Eastern Conference First Round and came back to win the series, winning Games 5 and 7 on the road, each in overtime.
"I believe fully we've earned the right to play our best hockey in a difficult situation," Maurice said. "All we want to do is get this thing back here. It's not the series, it's not a game, it's just get this thing back home and give our fans another look at us."
Stephenson got the Golden Knights started by scoring 1:39 into the first period to make it 1-0.