The Capitals snapped their two-game scoreless drought, but not a two-game losing streak despite a tying goal from Brooks Laich with under three minutes left in regulation.
In overtime, Brandon Sutter picked up a loose puck along the right boards in the Washington zone and skated up ice, finding Faulk in front of two defenders in the slot for the winner at the 3:38 mark.
Jiri Tlusty, Chad LaRose and Sutter also scored for the Hurricanes, who won their third game in a row over the Capitals and improved to 2-1-3 in their last six overall.
Cam Ward made 46 saves, including three in overtime.
"You know they're going to get their shots," said Carolina coach Kirk Muller. "We just wanted to take their time and space away after their initial shots and make sure they didn't get their third and fourth rebounds.
"That's what you have to do because they're so talented and so skilled, if you give them opportunities, they're going to capitalize."
Troy Brouwer's goal 13:59 into the first period gave Washington a 1-0 lead and snapped long scoreless droughts for the Capitals, who were shut out in their two previous games coming in. They had also lost two straight to Carolina, both by shutouts.
Jay Beagle also scored for the Caps while Michal Neuvirth allowed four goals on 36 shots. The Caps are just 4-7-2 in their last 13 games and 1-2-1 on a five-game homestand that ends Thursday against Tampa Bay.
Laich tied the game on a tic-tac-toe play with Marcus Johansson, who took his teammate's pass on the right side and sent it back into the slot, where Laich tipped it in to make it 3-3 with 2:52 remaining in regulation.
Earlier, Brouwer was carrying the puck on the left side when his backhand on net didn't make it through, but he regained the puck and took it around the back to the low right side, where he flipped a backhand up and in.
The goal ended the Caps' overall scoreless streak at 133 minutes and 59 seconds since Alex Ovechkin's overtime winner against the Islanders last Tuesday. It snapped an even longer drought against Carolina at 152:47 since Dmitry Orlov's game-winner early in the third period of a 2-1 victory Jan. 15.
"We played hard enough (to win) but we made mistakes on the back end tonight and it cost us," said Capitals coach Dale Hunter.
Beagle's rebound goal 4:11 into the second period was followed by three goals in a row for Carolina, including Tlusty's tip-in 40 seconds later and LaRose's rebound at the 8:30 mark.
Sutter then gave the Hurricanes a 3-2 lead on a breakaway 1:44 into the third period after Ward made a stop at the other end and the puck was flipped out of the zone. Sutter deked to his backhand to beat Neuvirth for his 100th NHL point.
It was the final game of the season between the Southeast Division foes...The Hurricanes have won three straight in the series after losing eight of the previous nine meetings, including three in a row earlier this season...The Caps were shut out 5-0 by New Jersey last Friday and 1-0 by Philadelphia on Sunday in their last two games. They lost to Carolina 3-0 on Jan. 20 and 5-0 on Feb. 20...Caps defenseman Dmitry Orlov received a roughing minor and a 10- minute misconduct for abuse of an official after a series of scrums in the second period when shoved a referee while trying to get at a Hurricanes player...Carolina started a five-game road trip that continues Wednesday in Buffalo. It kicks off a tough stretch that will see the Hurricanes play 10 of 12 games on the road.