The Capitals swarm defenseman John Carlson after his game-winner in overtime against Arizona. (Toni L. Sandys/The Washington Post)

Just moments after the public-address announcer at Capital One Arena declared one minute left in overtime, Capitals winger T.J. Oshie whiffed on a shot from the high slot. The puck bounced to the goal line, where defenseman John Carlson just happened to be waiting. He fired Washington’s 40th shot of the game and his 50th shot of the season, and the team and the player were simultaneously rewarded for their patience.

After storming back from an early two-goal hole to force overtime, the Capitals beat the Arizona Coyotes, 3-2, on Monday to record their first three-game winning streak of the season thanks to Carlson’s first goal of the campaign. The NHL’s worst team gave Washington (8-6-1) all it could handle through 64-plus minutes but, considering the abundance of opportunities the Capitals had, this was a game the team deserved to win.

“I think we’re just kind of finding our way and figuring out what’s working for us,” Carlson said. “We’ve got to put a couple of these together. We didn’t start [the season] like we wanted to, so we’ve got a lot of work to do, but just keep finding ways to win.”

The Capitals have played just six games at home this season, and they have struggled in that small sample size. The power play has been especially miserable here, entering the game as the worst in the league on home ice, scoring on just 1 of 19 opportunities.

With Washington down 2-1 in the second period Monday, the team’s man-advantage got plenty of practice with the Coyotes playing undisciplined and gifting the Capitals one power play after another. On Washington’s fourth power play of the period, Mario Kempe’s stick broke and Alex Ovechkin capitalized with a signature one-timer from the left faceoff circle that made it a tie game entering intermission. The goal was Ovechkin’s 12th of the season and his second in as many games. It also snapped a six-game drought for the Capitals’ power play.

“Sometimes it’s working, sometimes it’s not,” Ovechkin said. “Finally it’s nice to get a goal. It was a big goal for us.”

Before the game, Capitals Coach Barry Trotz warned that this could be a “trap game.” Arizona entered the matchup having not won a game in regulation this season, and with Washington playing the Sabres in Buffalo on Tuesday night and the Pittsburgh Penguins on Friday night, it would have been easy to overlook the Coyotes. But Trotz cautioned that while Arizona’s record may not be impressive, some of the team’s young talent is.

Any notion that the Capitals should not take the Coyotes seriously went away in the first minute of the game. Lars Eller’s shot went off rookie Clayton Keller, who raced down the ice with the puck. On a two-on-one, Keller scored his 11th goal of the season, bolstering his case as an early favorite for NHL rookie of the year. Less than six minutes later, Arizona made it 2-0 when Christian Dvorak’s centering pass was deflected in by Christian Fischer.

That gave the Coyotes an efficient two goals on their first two shots. They didn’t get their third shot of the game until there was 7:40 left in the first period, and Capitals goaltender Braden Holtby made his first save of the night, stopping Brad Richardson’s attempt in front. Had Holtby not made that save, Trotz admitted he probably would have pulled him at that point.

“I felt like he was going to mix in a save or two, and I wanted him to fight through that,” Trotz said. “. . . It’s a trust thing. They got two, and I got a couple looks on the bench, ‘Are you going to pull him or not?’ I said, ‘No, he’s going to mix in a save here and let him battle through this.’ ”

Two minutes after Fischer’s goal, Capitals forward Devante Smith-Pelly cut the deficit in half. Eller’s shot was blocked, and Smith-Pelly collected the rebound, backhanding it into the net, as the Capitals began to navigate their way to a win.

“You’re sitting there, you’re playing pretty well, and you’re down 2-0,” Trotz said. “I’m looking up and I’m going, ‘Just stay the course.’ I thought we did. Sometimes when you give up two goals as quickly as we did with the first two shots that they had, sometimes you get away from how you have to play. I don’t think we did. . . . That was the thing that impressed me the most — that there was no panic on our bench.”