Justin Faulk and Anthony Stewart scored in an 88-second span and Andreas Nodl added a goal during Carolina's three-goal first period.
Peters needed to make just 17 saves to win his second straight start. He replaced Cam Ward, who was out with an undisclosed lower body injury.
Jiri Tlusty had two assists, giving him points in a team-best eight straight games. The Hurricanes led 4-0 after 40 minutes and won their third in four games, improving to 11-3-1 in their last 15 at home.
Tomas Vokoun allowed two goals on seven shots before he was pulled at 5:09. Michal Neuvirth made 30 saves but gave up the final three goals.
The Capitals have lost five of six, and in two visits to Carolina since coach Dale Hunter took over for Bruce Boudreau, they have been outscored 8-0.
Meanwhile, Staal is playing his best hockey of the season, and it's not a coincidence that the Hurricanes are, too.
After spending most of the season in last place, Carolina has pulled even with Buffalo for 14th place in the Eastern Conference. The Hurricanes are one point behind fourth-place Tampa Bay in the Southeast Division.
Staal has six goals and five assists during his current six-game scoring binge. His first goal came 43 seconds into an elbowing minor to teammate Derek Joslin while the Capitals were trying to move the puck on their power play.
When diving defenseman Dennis Wideman couldn't get to a pass at the blue line, Staal chased the puck down and beat Neuvirth on a short-handed breakaway to put Carolina up 4-0 with 14:04 left in the second.
His second goal came with 14:09 left, when he scored on a snap shot from the right circle.
But by that point, this one had already been decided after the Hurricanes took a 2-0 lead 5:09 in.
Tim Brent uncorked a slap shot on a power play, and the long rebound sailed into the right circle. Faulk was there to slide it under Vokoun's pads at 3:39, giving the 19-year-old defenseman power-play goals in consecutive home games.
Stewart made it a two-goal game 1:28 later with a goal in transition, taking a feed from Staal and burying a shot. Nodl stretched it to 3-0 with 29.6 seconds left in the period when he one-timed a pass past Neuvirth.
NOTES: Washington didn't register its 10th shot until there were roughly 4½ minutes left in the second. ... Capitals C Matt Hendricks and Joslin fought twice in a span of 2:04 of the second. They each received 2-minute roughing penalties after the first fight — then brawled again immediately after they left the penalty box. ... Thirteen of the teams' previous 18 meetings were decided by one goal.