Alex Ovechkin’s go-ahead goal in the third period helped the Capitals to a 3-2 win over Arizona. (Toni L. Sandys/The Washington Post)

There were moments in the past 13-plus months that Mike Richards wondered whether he would ever score an NHL goal again, but it seemed inevitable on Monday night. One oh-so-close chance followed another, and his first goal as a Washington Capital finally came on a fortunate break near the midpoint of the third period. Tom Wilson dove for a puck that Richards was sure Arizona Coyotes goaltender Louis Domingue had covered, but it squirted out to the front of the crease, and Richards sent it home.

“I just figured with my luck it’d be disallowed,” he said.

He raised both arms before triumphantly slamming down his right fist. Along the bench, his teammates happily tapped him on the head. After signing with the team on Jan. 6, Richards went 14 games without a point, contributing in other ways but not on the score sheet. When his goal was announced, he received a standing ovation for a tally that proved the game-winner in Washington’s 3-2 victory over the Coyotes.

“He’s certainly got a reputation for scoring goals and being an offensive guy,” said Justin Williams, who also played with Richards on the Los Angeles Kings. “He’s a great two-way complement on our team, and he’ll be a big part going forward.”

The circumstances that led up to the goal started with an illness: Richards was only centering the third line because the Capitals were without third-line center Marcus Johansson, who missed the morning skate because he was sick. Johansson arrived at Verizon Center and got an IV before the game, but he ultimately told Capitals Coach Barry Trotz that he didn’t have enough strength to have a positive impact.

When Johansson missed four games with an upper-body injury after the all-star break, Richards had also centered wingers Jason Chimera and Tom Wilson on the third line, but the trio didn’t score a goal with him in it. Against the Coyotes, who allow the most goals per game in the league, the line had plenty of chances. Even before the goal, it had clearly been Richards’s best game since he joined the Capitals, as he recorded four shots on goal in the first two periods before finally breaking through .

“I was frustrated,” Richards said. “It is what it is. There’s not really much you can do about it except just keep going.”

Said Chimera: “We kept on saying that we were going to get one for sure. Our line was going pretty good, and we had some chances in the first and some chances in the second. We wanted to keep going and had a little frustration set in, but it’s nice to get that one for sure.”

Richards’s goal came just 26 seconds after Alex Ovechkin’s 39th of the season gave the Capitals their first lead 8 minutes 17 seconds into the third period. The insurance proved necessary when the Coyotes responded with a goal by Connor Murphy exactly a minute after Richards had found the net.

The Coyotes had taken a 1-0 lead 3:25 into the second period on a goal by Kevin Connauton. Evgeny Kuznetsov then responded on a power play with his 18th goal of the year 6:03 into the second.

Though Richards was frustrated with not producing points like he had earlier in his career, he had carved out a different role on the team, mostly centering the fourth line. Richards was good on faceoffs and a top penalty-killer, and Trotz trusted him so much defensively that he recently started playing Richards on a top line with Nicklas Backstrom and T.J. Oshie when Washington was protecting a one-goal lead late in games.

“Anytime you come to a team, I know you want to start getting a couple goals or assists or something,” Trotz said. “. . . He’s seeing how much we value him. I think he also sees how much his teammates value him. He’s proud, so he wanted to get on the board. There’s no question.”

A scrum broke out immediately after Richards scored his goal, leaving him to celebrate by himself before he noticed the commotion behind the net. Still, even in the chaos, Chimera fetched the puck so that Richards could have a keepsake. He had thought his first tally was coming, but finally getting it was a relief.

“I’ve also thought that I might not score ever again,” Richards said. “I’ve started feeling more comfortable probably the last half-dozen games or so, making more plays and not just panicking with the puck when you get it.”