Dwight King also scored for the Kings while Jonathan Quick made 27 saves.
Jeff Skinner had Carolina's goal, and Cam Ward stopped 36 shots.
Carter, who had scored four goals in his first four games of the season, went to his backhand in the shootout to beat Ward.
Quick stopped Nathan Gerbe, Alexander Semin and Skinner in the shootout to make Carter's goal hold up as the winner.
"He's a top-class goalie, and he showed it," King said of Quick.
Ward, Carolina's No. 1 netminder, has yet to win a game this season both of the Hurricanes' victories have come with backup Anton Khudobin in net.
Canucks' Edler suspended for check
Vancouver Canucks defenseman Alex Edler has been suspended three games for an illegal check to the head of San Jose Sharks forward Tomas Hertl.
The NHL made the announcement on Friday. The hit came late in the second period of Vancouver's 4-1 loss to San Jose on Thursday night.
Edler popped Hertl's helmet off as the 19-year-old was reaching for the puck. There was no penalty called on the play.
Edler was suspended for charging in March and is considered a repeat offender. He'll forfeit $182,926 in salary, which goes to the Players' Emergency Assistance Fund.