Justin Faulk and Riley Nash also scored for Carolina, winners of its past two.
Mike Fisher and Viktor Stalberg scored for Nashville, which completed a four-game homestand 0-3-1.
Faulk scored the game's first goal just 23 seconds after the opening faceoff with a wrist shot from the right faceoff dot that beat Nashville goaltender Marek Mazanec between the pads.
Nashville tied it at 50 seconds of the second on Fisher's goal.
Defenseman Victor Bartley fired a slap shot toward the Carolina net from the left point. Standing just to the left of the slot, Fisher tipped the puck over the shoulder of Carolina goaltender Justin Peters.
Bartley's assist was his first point of the season.
Less than two minutes later, Stalberg gave Nashville the lead at 2:41.
Coming down the right side on a 2-on-1 break with Nick Spaling, Stalberg's initial shot was turned aside by Peters, but he skated around the goal and swatted it into the net with a backhand for his third goal of the season.
Skinner evened the game at 2 at 6:11 of the second on a wrist shot from the right circle low to Mazanec's glove side.
Carolina regained the lead at 11:51 of the second.
Just out of the penalty box, Ron Hainsey intercepted a clearing attempt by Nashville's Roman Josi leading to a 3-on-1 break for the Hurricanes. Hainsey found Patrick Dwyer, who made a nice pass to Nash for his second of the season.
Nash's score snapped a 19-game goal-scoring drought.
Skinner's second goal came on the power play at 19:08 of the second when he ripped a slap shot from above the circle by Mazanec.
Carter Hutton replaced Mazanec in the Nashville goal at the start of the third period. Mazanec stopped just nine of the 13 shots he faced in 40 minutes.
Skinner completed the hat trick at 17:53 of the third with an empty-net goal.
Carolina's Patrick Dwyer, Mike Komisarek, and Dryason Bowman all left the game with injuries and did not return.