The Hurricanes seized a 3-2 lead in the best-of-seven championship series and will try to close it out on the road in Las Vegas on Sunday.
"It required everything we have," Hurricanes captain Jordan Staal told broadcaster ESPN. "I thought everyone in the group all through the lineup was dialed in tonight.
"It required everything, and it's going to require everything to finish this off."
Staal scored for the fifth straight game, Sebastian Aho added a goal and Carolina goaltender Brandon Bussi made 22 saves in his second straight start - becoming the first goalie in NHL history to make his first two career playoff starts in the Stanley Cup Final and win both.
Pavel Dorofeyev scored twice for Vegas, his power play goal less than even minutes into the opening period putting the Golden Knights up 1-0.
Staal struck less than five minutes later, redirecting a Nikolaj Ehlers shot past Golden Knights goalie Carter Hart and into the net.
Staal became the fifth player to have a five-game goal streak in the Stanley Cup Final and just the fourth to score in five straight games to start a Cup Final.
Svechnikov gave Carolina a 2-1 lead with a power-play goal from the top of the left circle at 11:58 of the second period and Aho pushed the lead to 3-1 at 17:51.
Svechnikov's second power-play goal, from in front of the right post at 13:49 of the second made it 4-1.
But Dorofeyev clawed one back with his second goal of the night at 13:49 of the third.
That raised the tension at the Lenovo Center in Raleigh, North Carolina, where fans were well aware that the series had seen multi-goal deficits erased in each of the first four games.
But the Hurricanes held on to move one win away from a second Stanley Cup title to go with the one they won back in 2006.
The Golden Knights, who won their only Stanley Cup in 2023, were without William Karlsson for much of the contest after the center exited in the second period with an apparent arm injury.