Skinner had two goals and an assist, and Peters stopped 26 shots for the Hurricanes, who had lost two straight and six of eight before bouncing back to match their most lopsided victory of the season.
"Took the team too lightly, I think," Capitals defenseman Karl Alzner said. "We watched the video, saw how things were going, they've got injuries, all that kind of stuff. That's bad on us."
Skinner's two goals and three points both tied career highs. Now in his fourth NHL season, the 21-year-old Toronto native last scored two goals in a game in January 2013, against Buffalo.
Skinner was playing in his seventh game after missing the previous 11 with an upper-body injury.
"He got off to a great start and then he got hurt," Hurricanes coach Kirk Muller said. "Now he's starting to get in the groove again. He's a goal scorer, and if you give him a chance, he's going to score."
Being lucky doesn't hurt. Skinner gave Carolina a 1-0 lead at 10:21 of the first period on a shot from the right side that struck goaltender Braden Holtby's bent left leg and bounced into the net.
"I think sometimes things just go your way," Skinner said. "The first goal, I just sort of tried to put it on net and it found a way in. Sometimes, that's just the way it goes."
Skinner connected on a wrist shot from the left circle to make it 2-0 at 11:18 of the second period. Less than three minutes later, he got an assist on a power-play goal by Andrej Sekera.
"Skin's a really gifted player, and he goes right to the hard areas," Peters said. "Tonight he got rewarded from playing within our system. It's great to see. I'm happy for him."
After Sekera scored, Jiri Tlusty connected less than a minute later for a 4-0 lead.
Mike Green's long-awaited first goal of the season, with Washington on the power play at 6:39 of the third period, ruined Peters' bid for his fourth career shutout. Two of the goaltender's three NHL shutouts are against the Capitals.
"They've got a lot of firepower there," Peters said. "Green made a nice shot. I'm just happy I was able to stick with it and get the win."
Holtby was 4-2 with a 1.84 goals-against average in six career games against Carolina, but in this one, he was pulled before the third period after yielding four goals on 23 shots.
Two of Washington's seven defeats at home this season have come against the Hurricanes. The loss ended the Capitals' two-game winning streak.
"I think every team goes through their highs and lows," Green said. "We have been very successful at going through streaks in the past, and that's maybe why it seems like we are so up and down."
Washington C Brooks Laich was scratched with a groin injury. ... Former Capitals star Alexander Semin missed an eighth straight game for Carolina with a concussion. ... Carolina is 9-0-1 when allowing two goals or fewer. ... Washington has scored 14 power-play goals in its past 16 games. ... Since the start of the 2005-06 season, Carolina is 26-2-4 against Washington when scoring at least three goals.