Alex Ovechkin fires the puck past Islanders defenseman Travis Hamonic and into the net for the game-winner. (Toni L. Sandys/The Washington Post)

For 56 minutes Tuesday night, the Washington Capitals simply couldn’t find a way to put a puck past New York Islanders netminder Evgeni Nabokov. With time waning, they stood down two goals and risked falling behind in a race for the postseason where every set of points is precious.

But just when the game might have slipped away, the Capitals broke through.

Troy Brouwer, who entered the contest on a 13-game scoring drought, recorded a pair of goals in the final 3 minutes 29 seconds of regulation to force overtime. In extra time, Alex Ovechkin sealed a 3-2 win in the dramatic fashion only he can provide with a rocket of a shot between Nabokov’s legs.

By all accounts it was an important win, one that advanced Washington to 69 points, moved the team into the final playoff spot in the Eastern Conference and created the first three-game streak since mid-January. But it is also the type of victory that can galvanize a group in what will be a tight battle to reach the postseason.

“Maybe you look back 20 games from now and it’s the game that sways the season,” Brooks Laich said. “It’s nice to fight back and win hockey games. . . . You look at guys’ faces and it’s excitement, enjoyment, relief, accomplishment — there was everything in there. We were scrappy.”

Despite outshooting the Islanders for much of the first two periods of play, the Capitals entered the final frame trailing by one after an early goal by Josh Bailey 4:16 into the first period.

New York made it 2-0 early in the third, though. A point shot by Steve Staios redirected off Matt Hendricks’s skate, then the stick of Matt Moulson and knuckled its way into the upper right hand corner behind Michal Neuvirth (22 saves).

As it continued to press to crack Nabokov (32 saves), Washington took solace in the fact that its forecheck generated chances consistently throughout the contest. The Capitals were able to sustain offensive pressure and once New York dropped back into even more of a prevent defense with the two-goal edge, the message on the bench was to stay the course.

“We were saying on the bench, ‘Just keep pushing, keep pushing. Keep playing down in their end and something’s going to give,’ ” said Jay Beagle, who finished with a team-high five shots in just more than 11 minutes of play. “Either they were going to give us a power play late or we were going to score. We potted one with Brouwer’s goal there and the bench went crazy. We knew that we were coming back — it was a great feeling.”

It was a play when Washington’s pressure down low worked perfectly. Jason Chimera used his speed to beat an opponent back to the puck deep in the Islanders’ zone. The turnover sent the puck to Mathieu Perreault, who flung the puck toward the net. Whether Perreault knew Brouwer was standing in the crease is something the players joked about afterward, but either way the rangy winger managed to tip the puck home.

Brouwer’s tally snapped his lengthy personal drought and gave the Capitals all the hope they needed as he cut their deficit to 2-1, 16:31 into the third.

“I’ve been struggling a little bit lately,” Brouwer said. “It’s nice to break out in such dramatic fashion but two big points by the team is what’s most important.”

A little more than three minutes later, after a faceoff win by Jeff Halpern, Brouwer plunked down on top of the crease again and tipped a shot past Nabokov to make it 2-2 with 25.5 seconds left, ensuring that the Capitals would at least obtain a point. It sent the Verizon Center crowd roaring and as television cameras zoomed in on the Islanders’ bench they caught Coach Jack Capuano utter just one word: “Unbelievable.”

Adrenaline from the comeback carried over into the overtime session. On his first attempt at a shot through a screen in the extra time, Ovechkin threw the puck wide. After a poke check by Mike Green caused a turnover in the Washington zone and sprung Ovechkin on another rush up ice just seconds later, he wouldn’t miss.

On a one-on-one, Ovechkin used defenseman Travis Hamonic as a screen and rifled a shot between Nabokov’s legs.

“It’s a huge victory for us,” Ovechkin said. “I had a lucky goal. I got the screen and I think I tried probably three times today to get that shot through.”