Martin Necas gave Carolina its first lead when he made it 5-4 with a power-play goal at 13:58, but Cody Eakin tied the game 5-5 at 16:07.
Andrei Svechnikov scored in the second round of the shootout for the Hurricanes, and Justin Williams ended the game with his goal in the top of the third.
[WATCH: All Hurricanes vs. Golden Knights highlights]
Sebastian Aho scored his 30th of the season to extend his goal streak to five games, and defenseman Jake Gardiner had three assists for Carolina (32-20-3), which is 2-1-0 on its four-game road trip. James Reimer made 33 saves.
"There's no quit in this team," Aho said. "They played a really good first period, and we were kind of off, but I have to credit them because they were good. We went to work and didn't stop believing. You could feel it on the bench. That's a good feeling."
Video: CAR@VGK: Aho puts home loose puck for 30th goal
Erik Haula had a goal and an assist in his first game at Vegas since being traded to Carolina on June 26. He played the past two seasons with the Golden Knights after being selected from the Minnesota Wild in the 2017 NHL Expansion Draft.
"It was pretty crazy," Haula said. "Even when we came out for warmups and came out for the game, I wasn't surprised. It was the norm. It felt good."
Defenseman Shea Theodore had a goal and an assist for the Golden Knights (28-21-8), who led 3-1 and 4-2 in the third period. Marc-Andre Fleury allowed five goals on 34 shots, and Jonathan Marchessault and Theodore missed on their shootout attempts.
"Not a good game from us," Marchessault said. "Overall, it was just a bad game. Gave away too many chances, kind of played on our heels in the third period."
Video: CAR@VGK: Reimer denies Theodore late in overtime
It was Peter DeBoer's first home game as Vegas coach since replacing Gerard Gallant on Jan. 15. The Golden Knights, who played their previous eight games on the road, are 4-2-2 under DeBoer.
"Strange game," DeBoer said. "I was worried a little bit about the game coming off the road. I thought we really started well, I liked our first period. ... I thought the second and third we didn't manage the game well.
Max Pacioretty gave Vegas a 1-0 lead at 13:26 of the first period when he redirected a Theodore shot.
Video: CAR@VGK: Pacioretty tips in Theodore's shot
Jon Merrill scored with a backhand from the slot at 14:35 to make it 2-0 after William Carrier brought the puck into the Carolina zone on the left side, skated it behind the net, and passed to the defenseman in the right face-off circle.
Teuvo Teravainen, playing in his 300th game in the NHL, scored with a wrist shot through traffic from the right point to make it 2-1 with 29 seconds left in the first.
Theodore scored for the second consecutive game to give the Golden Knights a 3-1 lead 59 seconds into the second.
Aho scored for the sixth time during his streak to get the Hurricanes within 3-2 at 1:19 of the third, but Chandler Stephenson scored shorthanded at 2:17 to make it 4-2.
Haula scored on the power play to make it 4-3 at 4:16.
Haydn Fleury tied it 4-4 at 10:48. The goal initially was disallowed for goaltender interference, but Hurricanes coach Rod Brind'Amour challenged, and the call was overturned after video review.
"They pushed, and we turned pucks over and took some penalties," DeBoer said. "Some uncharacteristic things. We've been playing some real solid hockey for a while, and the second and third, I thought, we were our own worst enemy."
Video: CAR@VGK: Fleury scores on wrist shot from point
They said it
"I had a lot on my mind; I sat out the whole year and had a lot to focus on. There's a lot of work ahead of me still. I'm proud of where I got myself to, from where I was, and every night I'm trying to play my best. I only have good memories, except for that game [at the Toronto Maple Leafs on Nov. 6, 2018, when he sustained a season-ending leg injury]. I love a lot of those guys over there [with the Golden Knights]." -- Hurricanes forward Erik Haula
"There's no excuses for our group. It starts with a guy like me too, just not ready mentally, physically. It starts with me. If a guy like me doesn't show the example, it's just going to creep down into our game." -- Golden Knights forward Jonathan Marchessault
Need to know
The Hurricanes rallied from down two goals to win for the second consecutive game. They trailed 2-0 at the Arizona Coyotes on Thursday before winning 5-3. ... Aho has six goals and four assists during a six-game point streak. ... Svechnikov had an assist on Teravainen's goal to extend his point streak to four games (three goals, three assists). ... Marchessault had his point streak end at six games (three goals, four assists). ... Theodore has nine points (two goals, seven assists) in the past seven games and 11 (two goals, nine assists) in the past nine.
Hurricanes: At the Dallas Stars on Tuesday (8:30 p.m. ET; ESPN+, FS-SW, FS-CR, NHL.TV)
Golden Knights: At the Minnesota Wild on Tuesday (8 p.m. ET; FS-N, FS-WI, ATTSN-RM, NHL.TV)
Video: Hurricanes score four in third, win in shootout