Rangers goaltender Igor Shesterkin made 36 saves for a second Game 7 win this postseason.
"Obviously, our goaltender was once again our best player," said forward Chris Kreider, who scored twice in the victory.
It's the first trip to the Stanley Cup semifinals for the Rangers since 2015, when they also faced the Lightning.
The Eastern Conference finals begin Wednesday night at Madison Square Garden. The Rangers won all three meetings with the Lightning in the regular season, with Shesterkin in goal for each victory.
Monday's Game 7 ended a 13-game playoff streak for the Hurricanes in which the home team won every game, and it also marked the first time an opponent scored more than two goals in a game at PNC Arena.
"I've kind of referred to us as cockroaches. We just didn't go away. That's always been engrained in the culture of every good team I've been on here. We just don't go away, regardless of the score and regardless of where the game is. We just keep on trying to find our game and playing for each other," said Kreider.
Fox gave the Rangers the early 1-0 lead, as New York's power play scored for the fifth straight game. New York earned the power play after goalie Antti Raanta turned the puck over deep in his zone, leading to a Sebastian Aho hooking penalty. The Hurricanes had killed off most of that man advantage when Fox jumped off the Rangers bench. The Carolina penalty kill didn't pick him up. That allowed Fox to skate in, shoot the puck and beat Raanta glove side for the 1-0 advantage just 3:40 into the contest.
Carolina had a power play after that, failing to convert. The Hurricanes then took a bench minor for too many men on the ice at 7:46 of the first period; an injured Seth Jarvis couldn't leave the ice before a sixth skater hopped on, the result of another crushing hit from Rangers defenseman Jacob Trouba in the series. Kreider deflected a Mika Zibanejad shot past Raanta for the 2-0 lead just 14 seconds into the power play.
"Whenever you get the first goal, especially on the road, it helps. This crowd, when they get into it, they feed off that," said Fox. "It was even more important to follow up with a second one pretty soon after that."
Jarvis, a first-line winger, did not return to the game for the Hurricanes.
The Hurricanes' power play remained powerless in this series, failing to score on three opportunities in the first two periods.
As their offense struggled to find traction in the second period, the Hurricanes suffered another injury blow to a significant player: Raanta, who suffered a lower-body injury while doing a split on a Zibanejad scoring chance. Raanta started 13 games in place of injured starter Frederik Andersen, who hadn't appeared since April 16 with a lower-body injury. Raanta was helped off the ice by a trainer.
Rookie Pyotr Kochetkov entered the game and gave up a goal to Strome at 16:19 to make it 3-0 for the Rangers. Winger Artemi Panarin chipped the puck up the boards to Strome, who came in on a 2-on-1 with Copp. Strome opted to shoot and beat Kochetkov on the second shot he faced after replacing Raanta.
"I was still nervous. I felt a little better after that third goal in the second period," said Rangers coach Gerard Gallant. "They're too good a team. If you let your guard down, they can come back."
Carolina finally broke through on the power play in the third period, as Vincent Trocheck scored just its second power-play goal in 18 opportunities in the series. But the Rangers answered back 40 seconds later, as Chytil scored his fifth of the playoffs off a Carolina turnover.
Max Domi scored late in the third period for Carolina.
Shesterkin was great when the Rangers needed him, including a point-blank save on forward Jordan Martinook seconds into the third period to preserve the lead.
"He made some key saves at key times. He made that 2-0 lead in the first period stand up," said Gallant.
"We scored two quick goals and it helped me so much. I just tried to [do] my job, stop the puck," said Shesterkin.
The Rangers goalie was 3-0-0 with a .958 save percentage and a 1.30 goals against average against the Lightning in the regular season. The playoff series pits Shesterkin against his Russian countryman Andrei Vasilevskiy, winner of back-to-back Stanley Cups.
"He's the best goalie in the world right now. I think it'll be a good battle," said Shesterkin.