Kochetkov made 14 saves in the second period, including stopping forward Jesper Bratt with his right skate on a breakaway with 31 seconds remaining in the period.
"He stole the show for sure, 'Kuchie' was our best player by far," Carolina captain Jordan Staal said. "Not our best [as a team]. They created a lot of offense and got us on our heels a little bit."
Jesperi Kotkaniemi and Staal scored, and Jaccob Slavin had two assists for the Hurricanes (20-6-6), who have won six straight and are 10-0-2 during the point streak. Carolina moved two points ahead of New Jersey for first in the Metropolitan Division.
"We weren't very good, and they were good. I give them a lot of credit for doing what they wanted to do," Hurricanes coach Rod Brind'Amour said. "We were able to capitalize on a few of their mistakes. Our goalie obviously was great tonight."
Jack Hughes scored, and Bratt had an assist for the Devils (21-9-2), who have lost six in a row and ended a 12-game road point streak (11-1-1).
"Missed nets on some great opportunities, in on breakaways [when] we don't score," New Jersey coach Lindy Ruff said. "There was a lot of stuff I thought we did really well. We had about four breakdowns and three of them on goals. When we beat ourselves on those goals and you're not scoring, we needed to hang around the game a little bit tighter."