Jason Chimera scored for Washington, which ended its three-game homestand with a 2-0-1 record and are the only team in the Eastern Conference without a regulation loss this season.
Alex Ovechkin scored the decisive shootout goal; Nicklas Backstrom, who played his 500th game, and Evgeny Kuznetsov also scored in the breakaway competition. Justin Peters made 20 saves in his first start of the season.
"We could have been up three or four goals after two periods, we had enough chances but we weren't," Capitals coach Barry Trotz said. "We had to scratch and crawl to get the two points. That is another thing we can work on. We tried the comebacks, we've had a couple comebacks this year. We're doing a lot of good things."
Brad Boyes scored for Florida, which has five goals in five games. The loss extended the Panthers' winless streak at Verizon Center to nine games (0-7-2) and they are 6-15-5 against their former Southeast Division rival since the start of the 2009-10 season.
"We've got to find ways to score more [goals], to generate more offense," Boyes said. "We've got to sustain more pressure. [Washington] has a good team, but we've got to realize that so do we. We've got to start playing in sync a little bit more. I think when we do that, [we'll be a better team]. We got one [point]; points are always important, even now."
Al Montoya stopped 27 shots in his first start for Florida.
"He was great," Panthers coach Gerard Gallant said. "[Washington] got one goal by him. I thought he made a lot of key saves. He was excellent tonight. I thought he gave us a chance to get that point and he was huge for us."
Chimera broke a scoreless tie at 17:12 of the first period. Washington worked the puck down low, with Joel Ward chipping the puck to Eric Fehr in the right corner below the goal line. Fehr then sent a centering pass through two Panthers toward Chimera, who outwaited Montoya and deked to his backhand for his first goal of the season.
"Well, I'm a goal scorer," Chimera said jokingly. "[Fehr] just made a good play. I knew [Montoya would] have to come out pretty fast. I knew no one was on me, so I kind of showed some patience. Great pass by [Fehr]. All I had to do was finish it."
The Capitals seemingly took a 2-0 lead midway through the second period when Brooks Laich tipped in Mike Green's floating wrist shot from the right point, but it was quickly waved off with Laich's stick above the crossbar.
Two minutes later, Chimera roared up ice in search of his second goal and once again went to the backhand, but Montoya denied him with a sprawling left-toe save.
"He had a lot of energy. A little bit too much at some points," Fehr said of Chimera. "We had to talk to him, say 'Hey, calm down,' but that's the way he is. He skates hard, he plays hard and that's what I like about him."
The Panthers pushed back to begin the third period and Boyes tied the game on the power play at 4:27. Florida, which did not register a shot on goal during its previous three power-play opportunities, scored when Boyes put Sean Bergenheim's rebound past Peters.
Laich left the game with about five minutes left in regulation after a shoulder-to-shoulder collision with Panthers forward Shawn Thornton. Trotz said Laich, who missed 70 regular-season games over the past two seasons because of groin injuries, will be evaluated Sunday.
"To be honest with you, I thought he hurt his knee because he blocked a shot right before that too," Trotz said. "He had hurt his knee blocking a shot two games previous and I thought it just hit him on the same spot. When he went off, someone told me it was upper-body, not lower-body.”
It was a rare low-scoring victory for the Capitals; Washington had not won a regular-season game when scoring two or fewer goals since a 2-1 shootout victory against the New York Islanders on April 4, 2013.
"It felt really good," Backstrom said. "I think a big part of it was [Peters] playing really good behind us. You've got to win these games. These points are really important at the end."
After playing four of their first five games at Verizon Center, the Capitals will embark on a three-game road trip against the Edmonton Oilers, Calgary Flames and Vancouver Canucks next week.
"It's a good springboard going into that," Capitals defenseman Brooks Orpik said. "[It's] always tough going out West. We had the luxury of starting four out of five at home and took advantage of it."