The Islanders were down a player on the bench, down a player on the ice and fighting through what Cal Clutterbuck called an “unplayable” ice surface in the third period at Barclays Center on Friday night.

The Islanders were desperate for a win in a tie game. Johnny Boychuk, one of the five healthy bodies on defense left after Ryan Pulock exited the game in the first period with a potentially serious leg injury, came to the rescue. Boychuk’s shorthanded one-timer off Brock Nelson’s faceoff win 4:24 into the third broke a tie and sent the Isles to a 3-2 win over the Coyotes, the Islanders’ first regulation victory of this young season.

“We just had to keep the shifts short and make simple plays,” said Boychuk, who recorded the first shorthanded goal of his career. “The ice was real bouncy today so there wasn’t much to do but throw it on net.”

The Isles’ defense corps had produced just one goal in the first four games and needed to give more, even with the absence of Nick Leddy, who is day-to-day with an upper-body injury.

Pulock was called up from Bridgeport on Thursday, ready to take Leddy’s spot on the power play and try to infuse some energy into the team. He got five shifts, including one power-play shift, before departing midway through the first period.

“It seems like we can’t catch a break,” Jack Capuano said. “We’ll know more tomorrow, but he could be out a while.”

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The Islanders got contributions from their defense, including Pulock, who started a play out of the Isles’ zone that was finished off with the opening goal at 4:45 by Dennis Seidenberg off a nice feed from Casey Cizikas.

Boychuk zipped through the Coyotes defense and threw a backhand on net later in the first. Coyotes goaltender Louis Domingue stopped it but couldn’t find the rebound at his feet; Ryan Strome did, giving the Isles a 2-0 lead on a special night for Strome, playing against younger brother Dylan with lots of Stromes in attendance.

“I’m not sure if I was his man or not,” Ryan Strome said, “but I’ve been catching the minuses a lot lately so it’s OK if [his brother] caught one there.”

Just 51 seconds later, the game was tied. A brutal line change allowed Brad Richardson to walk in and beat Jaroslav Halak at 14:09; 13 seconds later Radim Vrbata redirected Jakob Chychrun’s point shot and it was all even.

“I was really happy with the way we responded in the second,” said Halak, whose team outshot Arizona 16-4 in the middle period and 31-25 for the game. “We were shorthanded on the bench and all the guys stepped up.”

After Boychuk’s ripper the Coyotes pressed, but the Islanders were the stronger team. With the Arizona net empty the visitors didn’t get a shot on Halak in the final minute, with Cizikas strong-arming Anthony Duclair into the corner to run out the clock and give the Isles a well-earned win on a steamy night, buoyed by another perfect night of penalty killing with the shorthanded winner to cap it off.

“From about the 5-6 minute mark of the second, you knew it was one of those nights,” Clutterbuck said. “You basically couldn’t string three passes together, the ice was unplayable. But we found a way.”