Nico Hischier and Dawson Mercer scored, and Jacob Markstrom made 25 saves for the Devils, who are the No. 3 seed from the Metropolitan Division.
New Jersey trails the best-of-7 series 2-1. Game 4 will be here on Sunday (3:30 p.m. ET; FDSNSO, MAX, truTV, MSGSN, TBS, SN360, TVAS).
"Obviously, feels good," Hischier said. "I think we really [dug] in deep tonight and came out with a win, a big win for us. And now it's all about recover and do it again (in Game 4 on Sunday)."
Sebastian Aho had a goal and an assist, and Andersen made 34 saves for the Hurricanes, who are the No. 2 seed from the Metropolitan.
"I mean, we did fight back, tie it up," Hurricanes coach Rod Brind'Amour said. "That was an eight minute stretch where we were kind of good. The rest was not our game. You know, there's a reason they were good. So, it's tip your hat to New Jersey. They did what they had to do. We were just not very good."
Hischier gave the Devils a 1-0 lead at 16:11 of the first, scoring from in close after taking a lead pass from Timo Meier, who was on the right boards.
Andersen kept the game 1-0 by making a highlight-reel glove save on Meier during a net-mouth scramble at 13:56 of the second period.
"Freddie obviously kept us in that game, really," Hurricanes captain Jordan Staal said.
Markstrom made a left toe save on Aho, who had a short-handed chance at 2:25 of the second period, to preserve the lead for the Devils.
Mercer then increased the lead to 2-0 at 1:18 of the third with a wraparound at the side of the net.
"It [stinks] but you know, it's hockey and we've all been here. We got better and starting next game, we expect to come out better," Carolina defenseman Jalen Chatfield said.
Seth Jarvis cut it to 2-1 at 6:11 of the third with a power-play goal.
Aho tied it 2-2 with another power-play goal at 12:20. Andrei Svechnikov got a rebound and fed the puck to Aho, who scored from in front with Markstrom out of position.
The Devils played the final four periods with five defensemen after Johnathan Kovacevic sustained an undisclosed injury in the first period. They were also without defensemen Brenden Dillon and Luke Hughes, who were each out with an undisclosed injury. Jonas Siegenthaler was playing in his first game since Feb. 4 after being out with a lower-body injury.
"I'm just so impressed with our team tonight," Devils coach Sheldon Keefe said. "The defense in particular. So fitting that Nemec gets the game-winner for us. Our whole team, just the way we dug in tonight, stayed with it."