Roman Josi had three assists, and Saros made 25 saves for the Predators (28-18-4), who have lost four in a row.
"There's times and places to be able to make plays," Nashville coach John Hynes said, "but when you don't have speed and you don't have support and you don't have numbers on the puck, and you're getting checked the way that we were getting checked, we weren't able to get to the game that we wanted to regularly. So those are some areas we'll address."
Skjei gave the Hurricanes a 1-0 lead at 10:24 of the first period when he took a touch pass from Necas near the left wall following a face-off, drove to the net and scored on a backhand off the right post.
"I just made a read and found some open ice," Skjei said. "I took it to the backhand, which I think was the right play, and then put it in."
Skjei has scored seven goals in 48 games this season, one short of his NHL career high. He scored eight in 2018-19 (78 games) and 2019-20 (67 games).
"Because he can skate so well, he has the ability to get up in the rush," Brind'Amour said of the defenseman. "It's skating that got him that goal. He got down the wall and made a nice little move. He's been good all year. It's nice for us to see him score but it's not surprising."
Jesperi Kotkaniemi made it 2-0 at 5:14 of the second period when he scored in the slot on the rebound of his own shot.
Luke Kunin cut it to 2-1 with 1:22 left. He took a pass from Ryan Johansen in transition and scored on a snap shot from the high slot.
Slavin pushed the lead to 3-1 with one second left. He scored from the left face-off circle off a pass across the slot from Svechnikov.
"In the second period, we didn't play the game that gives you the best chance to win regularly," Hynes said. "We were trying to make plays when plays weren't there. They had a lot of juice and they wound up scoring late in that period, but they also had lots of other opportunities."
Svechnikov made it 4-1 on the power play at 6:43 of the third when he took a pass from Teuvo Teravainen and scored from the right circle. Svechnikov has scored nine points (six goals, three assists) in his past five games.
"It's awesome when you're doing those things," Svechnikov said. "I try to focus on the hard work. That's what it's all about. And if I'm going to be rewarded, I will be happy."