It was the first NHL win for Nedeljkovic, who played in one previous game, making the save on all 17 shots in relief during a 4-1 loss to the Columbus Blue Jackets on Jan. 17, 2017.
There were no signs of nerves from the 23-year-old goalie, who was selected by Carolina in the second round (No. 37) of the 2014 NHL Draft.
[WATCH: All Hurricanes vs. Canucks highlights]
"One of my goalie coaches told me growing up, 'Play every game like it's the biggest game of your life, so when you play the biggest game of your life, it's just another game,' so that's just kind of how I looked at today," Nedeljkovic said. "Just another Wednesday night game."
Nino Niederreiter scored twice, Teuvo Teravainen had a goal and two assists, and Dougie Hamilton and Greg McKegg each had a goal and an assist for Carolina (24-20-6), which went 2-0-1 on a three-game Western Canada road trip and is 9-3-1 in its past 13 games entering 2019 Honda NHL All-Star Weekend. Sebastian Aho had two assists.
Video: CAR@VAN: Niederreiter pots Williams' pass for PPG
The Hurricanes had 11 healthy forwards after Micheal Ferland sustained an upper-body injury in a 3-2 overtime loss to the Calgary Flames on Tuesday.
It was Carolina's first win in Vancouver since Oct. 15, 1999. They were 0-7-3 since.
"We played last night and it's the last game before we get a break, but they dug in," coach Rod Brind'Amour said. "We haven't had a lot of success here, so tonight was nice."
Josh Leivo and Sven Baertschi scored, and Jacob Markstrom made 29 saves for the Canucks (23-22-6), who went 3-1-2 on a six-game homestand.
"We've got to be a lot better coming back from the break if we're going to make a run for it," Vancouver center Bo Horvat said.
All seven goals in the game were scored in the second period.
Video: CAR@VAN: Niederreiter scores his second of the period
The Hurricanes went ahead 1-0 on the power play after a deflection caught Markstrom moving and left Niederreiter with an open net at 1:54.
Leivo tied it 1-1 with a redirection at 2:57, and Baertschi finished off a three-way passing play with Brock Boeser and Elias Pettersson at 5:14 to put Vancouver ahead 2-1.
"We gave ourselves a chance there in the second period, and right after that, we just kind of took our foot off the gas and just gave up way too many Grade A opportunities in our slot," Horvat said. "[Markstrom] didn't have a chance."
McKegg tied it 2-2 at 6:53, one-timing a pass from Saku Maenalanen from below the goal line, and Niederreiter scored his second goal on a pass from Aho from below the goal line to make it 3-2 at 13:17.
Niederreiter has four goals in four games with Carolina since being acquired from the Minnesota Wild in a trade for Victor Rask on Jan. 17.
"It's definitely a lot to take in but, at the same time, I know how good Aho is and [Justin] Williams is," Niederreiter said of his new linemates. "They are great playmakers, and for me, it's just trying to find the open ice and like tonight a terrific pass by Aho and I had a chance to do the rest."
Video: CAR@VAN: Baertschi finishes tic-tac-toe passing
Hamilton scored on another pass from behind the net 25 seconds later to make it 4-2 at 13:42, and Teravainen finished a 4-on-2 rush to make it 5-2 with 3:28 left in the period.
"They wanted these two points more than we did," Markstrom said.
They said it
"He can score. We've been looking for goals for a while. We play a really good style of play, but we need some finishers, and it was a great trade for us to bring a guy that has the pedigree to put the puck in the net." -- Hurricanes coach Rod Brind'Amour on forward Nino Niederreiter
"I thought we struggled against their quickness tonight. They're a quick team, and we knew it. And we struggled against it." -- Canucks coach Travis Green
Need to know
Aho has 12 points (six goals, six assists) in his past seven games. … It was the eighth time in the past 12 games the Hurricanes scored at least four goals. … There was no update on Ferland, who also missed a 7-4 win at the Edmonton Oilers on Sunday because of an upper-body injury. … Pettersson has a goal and two assists in two games since returning from a knee sprain. He leads the Canucks and NHL rookies in goals (23) and points (45).
Hurricanes: Host the Vegas Golden Knights on Feb. 1 (7:30 p.m. ET; FS-CR, ATTSN-RM, NHL.TV)
Canucks: At the Colorado Avalanche on Feb. 2 (10 p.m. ET; CBC, SN, SN360, SN1, CITY, ALT2, NHL.TV)
Video: Niederreiter scores two, Nedeljkovic earns first win