Eric Staal had two goals and an assist, Justin 0, on Monday night.
Justin Faulk and Anthony Stewart scored in first 5:09 of the game to give the Hurricanes a 2-0 lead and chase starting Capitals goaltender Tomas Vokoun.
Carolina opened a six-game homestand with its third win in four games. Andreas Nodl also had a goal.
Peters, seeing time in place of injured starter Cam Ward, picked up his first NHL shutout in his 26th game, including five this season. He had 17 saves.
"It feels great," said Peters. "It was a complete effort by everyone tonight and I'm going to enjoy it."
Vokoun gave up two goals on seven shots and was replaced by Michal Neuvirth, who made 30 saves on 33 shots the rest of the way. The Capitals lost for the fifth time in their last six games and fell to 4-7-3 in their last 14.
Faulk scored 12 seconds after Roman Hamrlik went off on an elbowing penalty, beating Vokoun on a rebound from the left side just 3:41 into the game.
The second goal was hardly Vokoun's fault, but he was taken out anyway after a Caps turnover at the blue line led to Stewart's goal from the slot to make it 2-0 Carolina.
Later in the period, Nodl gave the Hurricanes a 3-0 lead at the end of a 3- on-2 rush, beating Neuvirth with a one-timer from near the right circle after Brandon Sutter's cross-ice pass.
Staal's shorthanded goal came after Caps defenseman Dennis Wideman fell down at the blue line on a pass to the point. The Hurricanes captain hustled to the puck and raced down ice alone on Neuvirth, deking to his backhand and beating him through the five-hole.
"The building was full and there was a lot of excitement," said Staal. "The guys were ready to go and we played a full 60 minutes."
The game out of hand, Washington's Matt Hendricks and Carolina's Derek Joslin squared off to fight. They fell down, drawing roughing minors instead, but got the fight out of the way as soon as they came out of the penalty box.
Carolina's Justin Faulk and Washington's Troy Brouwer fought later in the period in a scrap initiated by Brouwer in the corner.
"Normally you can try to take some positives out of a game. Tonight, nothing," said Brouwer. "You can't wipe it clean because you have to take it in and make sure you know what you're doing wrong, make sure you know what you did wrong. We're going to watch a lot of video, I hope."
Staal scored again in the third period after a tired Washington unit gave the puck away in its zone and it ended up in front of him near the bottom of the right circle.
Carolina has won two straight over Washington after losing the previous three meetings. The Capitals have won eight of the last 11 matchups...The Hurricanes called up Mike Murphy from their AHL affiliate in Charlotte to back up Peters on Monday. Ward is day-to-day with a lower body injury...Both NHL games Monday ended in lopsided shutouts. Ottawa defeated the New York Islanders, 6-0...The Caps are 1-2 on a four-game road trip that ends Wednesday at Pittsburgh.