The 18-year-old forward also set up Carolina's first goal from Tuomo Ruutu, which tied the game two minutes into the second period. The Wild had opened the scoring 17 minutes into the first period on a goal from Andrew Brunette.
Cam Ward made 41 saves in the Carolina goal, stopping Mikko Koivu, Brent Burns and Antti Miettinen in the shootout to preserve the victory.
"This was my second game, and a great way to finish that off," Skinner said of his shootout goal. "First I was looking to shoot, but the puck was rolling on me and I tried to settle it down. It sort of opened up for me then.
"It's pretty big," Skinner added. "It has not really sunk in yet. So far I have a lot of memories from my first two games."
Minnesota captain Mikko Koivu led his team with a strong showing in the offensive zone, and his shot led to the first goal scored by Brunette on a rebound at 17:23 in the first period.
One of the keys for Carolina was its short-handed play for 5 1/2 minutes during the second period, including a 30-second stretch where it was two men short. Another was the goaltending of Ward, who was unruffled by crowding in the crease and covered every point-blank shot.
Minnesota had chances in the third period, with Martin Havlat getting two breakaways but failing to score, the puck gliding almost along the goalline on the second try in overtime.
The game Friday night reached a new level of intensity compared with the night before, when Carolina won 4-3. A pair of early fights and a nasty altercation between Minnesota's Eric Nystrom and Tom Kostopoulos of Carolina appealed to the 13,465 in the sold-out arena.
On the basis of the first two games, the swift, 5-foot-10 Skinner, with well-timed forays to the net, could stick around for Carolina. The team has seven games to decide whether to keep him or return him to the juniors.
"I hope to stay," Skinner said, "so I'll work hard and take each game as it comes and be better every day."
He was a bit intimidated by stepping into the NHL atmosphere at just 18 years old.
"But more by my first practice with guys as Cam Ward and Eric Staal than by the level of play itself. It was cool to be there with them. Just now I feel pretty comfortable," he said. "I got two great players, Jussi Jokinen and Tuomo Ruutu, who make the game a lot easier for me."