Brooks Laich triggers a celebration at Verizon Center after sending the Capitals — and new Coach Dale Hunter — to their first victory since the firing of Bruce Boudreau. (Rob Carr/Getty Images)

With each game there are signs of the defensive improvement Coach Dale Hunter is looking for from the Washington Capitals. Hunter’s message has remained constant: Through defense, offense will follow. Saturday night at Verizon Center, Washington finally saw that progression.

Brooks Laich, playing in his 500th NHL game, scored the game-winner just 12 seconds into overtime as the Capitals captured a 3-2 victory over the Ottawa Senators, snapping a four-game losing streak to give Hunter his first win as an NHL head coach.

“It feels good,” Hunter said of the first win. “It’s one of those things that you want to get over with right away, hopefully the first game, but it ended up being the third. It’s one of those things you want to get under your belt and move on.”

Against Ottawa, the Capitals’ style of play under Hunter took another step forward as the players started to look comfortable and confident with the new system.

Their transition game was cleaner and more fluid, they limited the Senators largely to shots from the outside and didn’t give up many odd-man rushes. And for the first time in the past three games, the Capitals’ offense dominated play for prolonged stretches. For Washington, to see the progress result in a victory is a confidence-builder.

“It takes a little bit. You’re drilled from training camp. Me and the guys who have been here a couple years, you’re drilled a certain way for a long time,” veteran Mike Knuble said. “You’re talking about changing a lot of habits in a couple days. We practice, and we’re not robots. It takes a couple times to get the feel of what you’re supposed to be doing and what everybody’s supposed to be doing on the ice.”

Washington (13-11-1) jumped out to a 1-0 lead just 1 minute 58 seconds into the contest when Nicklas Backstrom skated into the offensive zone, boxed in by four Senators, and took a wrist shot in the slot that beat Craig Anderson (32 saves). It was Backstrom’s ninth goal of the season and marked his 27th point, a team high.

Washington wouldn’t be able to give itself a larger cushion before a point shot by Senators blue-liner Erik Karlsson made it 1-1 only 58 seconds into the second. Karlsson’s shot may have been redirected on its way to the net where it fooled Michal Neuvirth, who finished with 32 saves in a strong outing for his first win since Nov. 4 at Carolina.

From the moment the Senators tied the score, though, the second period was Washington’s. Strong shifts by every line, particularly the third and first, yielded scoring chances in succession.

“We wanted to limit their shots and get more shots ourselves. And our line, especially, we had a couple hornets’ nests around their net where we had second and third whacks at it,” said Laich, who finished with a game-high six shots on goal in 20:45 of ice time. “The first couple games we kept getting one shot, but we weren’t getting to the rebounds and creating those other opportunities. So we did little bit more of that tonight and we got rewarded for it and get Huntsy his first win.”

Alex Ovechkin had the best power-play chance of the night on a partial breakaway less than two minutes after the goal by Ottawa (12-11-3).

A ruckus in front led to an excellent opportunity for Laich in the crease, but the center didn’t elevate the shot, which found a home in Anderson’s pad. A backhander by Brouwer roughly eight minutes in missed the net entirely but was such a golden chance that even Ovechkin already had his arms raised, poised for a celebration that wouldn’t come.

With each puck Anderson found, and each time the Capitals couldn’t finish after a shift of strong forechecking and possession deep in the offensive zone, it would have been easy for them to become discouraged. But Washington churned forward in its new system, grinding along until something gave.

“We had a lot of good chances, I think, and we should have put a couple more in there, but he was playing good in the net,” Backstrom said of Anderson. “You’ve got to keep going. It’s a 60-minute game and you’ve got to keep going and today was positive.”

In the third period, the Capitals finally added a finishing touch. Dennis Wideman jumped up in the play, carrying the puck into the offensive zone to help create a three-on-two, and passed to Ovechkin on the left side board. Washington’s captain sent a long pass across the ice to Brouwer, who sneaked backdoor for a goal that gave the Capitals a 2-1 edge 5:01 into the period.

Ottawa’s Milan Michalek scored on a rebound to even the score at 2, but despite the unplanned detour into overtime the Capitals kept pushing forward.

On the first shift of the extra frame, Jason Chimera raced up the left wing when Karlsson slipped and fell, creating a two-on-one for the speedster and Laich. Chimera passed to Laich in the crease and the center chipped the puck past Anderson.

“We’ve got out of a little funk with a win,” Laich said. “But we have to start stringing two, three, four, five together to make some progress that way.”

Capitals notes: Alexander Semin was scratched because of a right shoulder injury. Roman Hamrlik was out for the second consecutive game with an undisclosed lower-body injury. Mike Green missed his 11th consecutive game with a strained right groin muscle.