The Hurricanes return to Raleigh, North Carolina, for game five on Thursday at 2-2 in the best-of-seven championship series.
The Hurricanes had not lost two games in a row since mid-January and kept that run intact with a dominant third period.
Their 37-year-old captain Staal flipped a backhanded shot after falling over that went past Vegas goaltender Carter Hart to give the Hurricanes a 4-3 lead with 13:28 remaining.
It was Staal’s second goal of the game and his fifth of the final, more than any other player.
His first-period goal had already made Staal the oldest player to score in the first four games of a Stanley Cup Final.
He is the first to score five in the first four games since Pittsburgh legend Mario Lemieux in 1992.
“He’s a big time player, leads us every day, plays the right way and gets rewarded,” Carolina goaltender Brandon Bussi said.
Vegas pulled Hart for an extra attacker in the dying minutes, but Nikolaj Ehlers scored an empty-net goal for Carolina with 55 seconds remaining to seal victory.
Bussi made his first career playoff start as ‘Canes goalie Frederik Andersen was benched after surrendering four goals in the second period of game three.
“Kids grow up, they dream about doing something like this,” Bussi said.
“Excited for the win and to be going back to Raleigh.”
Carolina grabbed the lead on Logan Stankoven’s backhand goal only 66 seconds into the game, taking the puck after it rebounded off the board to the front of the net.
Unguarded Jackson Blake took a pass from Taylor Hall and fired in a shot from the right of Hart only 2:22 later for a 2-0 Carolina edge.
Mark Stone answered for Vegas with 7:22 elapsed on a breakaway goal, angling a shot past Bussi.
Staal batted in a rebound off Shayne Gostisbehere’s shot for a power play goal after 12:48 of the first period for a 3-1 Hurricanes lead.
William Karlsson blasted a shot past Bussi 4:22 into the second period to lift Vegas within a goal.
Brett Howden netted the equaliser on a two-on-one breakaway with 2:52 remaining in the period, his 14th goal of the playoffs.
It marked the first cup final with a multi-goal deficit erased in each of the first four games and set the stage for the closing drama.
The Golden Knights won their only Stanley Cup in 2023 while Carolina only took the trophy in 2006.