After suffering back-to-back shutout defeats on home ice, Washington went ahead 2-0 before Carolina rallied and grabbed the lead on a breakaway goal by Brandon Sutter at 1:45 of the third period.
The Capitals forced overtime when Brooks Laich scored off a centering pass from Marcus Johansson at 17:08 of the third, but Faulk took a centering pass from Sutter and fired a shot past Michal Neuvirth to hand Washington with its third straight loss.
The defeat left Washington -- which has qualified for the Stanley Cup Playoffs four straight years -- four points behind first-place Florida in the Southeast Division and two points in back of Winnipeg for the eighth and final playoff spot in the Eastern Conference.
Carolina scored off two odd-man rushes, received a gift-wrapped breakaway and benefited from a strange bounce off the back boards. It all added up to a rare road win for the Hurricanes and another frustrating night for the Capitals.
"Turnovers. Missed assignments. Just not playing the situation," Washington's Brooks Laich said. "If you're going to give up goals, you want to make them work for it."
Washington salvaged a point in the standings when Laich scored off a centering pass from Marcus Johansson at 17:08 of the third.
"It's not what we're hoping for, but it's better than nothing," Capitals right wing Troy Brouwer said of picking up a point in the standings.
It was the Hurricanes' eighth road win, still the fewest in the NHL.
"We felt like we played a pretty solid road game tonight and were deserving to get the two points," Carolina captain Eric Staal said.
Staal extended his point-scoring streak to 12 games with an assist on the Hurricanes' first goal. Staal's streak, during which he's registered seven goals and 14 assists, is the longest by a Carolina player since the team moved from Hartford before the 1997-98 season.
Brouwer notched his 18th goal of the season at 13:59 of the first period to end Washington's scoreless run at 133 minutes, 59 seconds. After firing a shot from the left circle, Brouwer plucked the puck off Ward's right skate, whisked around the net and put in a backhand from the right side for a 1-0 lead.
It was the Capitals' first goal against Carolina in three games, following shutout defeats of 3-0 on Jan. 20 and 5-0 on Feb. 20.
Beagle made it 2-0 at 4:11 of the second period by cashing in the rebound of Dmitry Orlov's shot from the left point. Beagle had gone 48 games since Dec. 28, 2010, without a goal.
"It seems like whenever they score, it's too easy," Beagle said. "We're giving them easy goals from either turnovers or missed coverage. I was guilty of a couple turnovers tonight, too."
"When you get down by two like that and make it a one-goal game again, it's huge, especially on the road," Sutter said. "It was a big momentum boost, and we managed to play pretty well from that point forward."
The Hurricanes tied it when Tim Gleason took a shot that bounced off the back boards to Chad LaRose, who beat Neuvirth from the left side at 8:30.
"It's one of them hockey plays that just happens, and I don't blame Neuvy on that one," Capitals coach Dale Hunter said.
Material from wire services and team media was used in this report.
1 - 0 WSH
2 - 0 WSH
2 - 1 WSH
2 - 2 Tie
3 - 2 CAR
3 - 3 Tie
4 - 3 CAR
Delaying Game-Puck over glass
Interference on goalkeeper