Andrei Svechnikov scored the only goal of the shootout while Raanta stopped Tomas Tatar, Jesper Bratt and Jack Hughes.
Bratt had two goals and an assist for the Devils (23-11-3), who have lost three of four and are 2-7-2 in their past 11. Mackenzie Blackwood made 43 saves.
"Those are two points I thought we could have had for sure," Blackwood said. "Just a tough finish to the game. We got one [point] but it's not what we were looking for."
Carolina held New Jersey to 28 shots on goal, including seven each in the first and third periods.
"It was tough to generate," Devils coach Lindy Ruff said. "Their gaps were incredible. A heck of a battle to get where we were, and at the end, we got a point. We'd like more."
Bratt tied it 2-2 at 6:15 of the third period when he beat Raanta five-hole off the rush from the left face-off circle.
Stepan put the Hurricanes back in front 3-2 at 8:04 on a rebound at the right post, but Nico Hischier tied it 3-3 at 8:38 on a wraparound after New Jersey won a puck battle in the corner.
"We gave some really uncharacteristic turnovers and things we don't normally do," Brind'Amour. "Against this team, you can't do that. They just came down and, boom. I like the resiliency though."
Bratt put the Devils ahead 4-3 at 13:48 when he scored off the rush with a wrist shot from the high slot.
"I felt like it wasn't a bad time to shoot," Bratt said. "Nico made a good screen and helped me out to make the goalie not see the puck. Luckily it went in."