Scott Gomez had a goal and an assist, and Jamie Langenbrunner added two assists for the Devils, who remained two points behind Buffalo in the race for first place in the Eastern Conference.
Martin Brodeur stopped 38 shots and beat the defending Stanley Cup champions for the third time this season.
David Tanabe and Andrew Ladd scored for the Hurricanes, who are tied with the New York Islanders for seventh in the East. John Grahame, making his second straight start after No. 1 goalie Cam Ward cut his leg Sunday against the New York Rangers, made 22 saves.
Rasmussen put New Jersey ahead to stay 40 seconds into the third, tapping the rebound of Gomez's hard shot past Grahame to make it 2-1.
Parise scored midway through the period when he took a centring pass from Travis Zajac and beat Grahame stick-side for his 28th goal.
Ladd made it 3-2 about three minutes later when he redirected Tim Gleason's long slap shot into the net. Grahame was pulled for an extra skater with about 90 seconds left, but Carolina couldn't generate any more offence against Brodeur.
The Devils took a 1-0 lead almost from the drop of the puck. Gomez scored 39 seconds in on the first shot of the game. He took a pass from Langenbrunner and scored with a wrister from just inside the right circle.
The Hurricanes controlled the rest of the opening period, with Tanabe tying it midway through the frame with his second goal of the season.
Carolina's first scoring play started when Ray Whitney intercepted the puck near centre ice, and passed to Justin Williams between the circles. Williams dropped the puck to Tanabe, who tipped it past Brodeur.
Notes: Brodeur entered having allowed a total of two goals in two victories over Carolina this season. ... The Devils improved to 3-0 in the season series. ... The teams meet again Saturday in New Jersey. ... Carolina D Bret Hedican (lower body injury) was scratched and is out indefinitely. ... Carolina captain Rod Brind'Amour's scoring streak ended at four games.