Devils right wing Kyle Palmieri misses a shot against Islanders...

Devils right wing Kyle Palmieri misses a shot against Islanders goaltender Thomas Greiss during the shootout on Thursday in Newark, N.J. Credit: AP/Julio Cortez

NEWARK — The overriding narrative in the Islanders’ season has been their marked defensive improvement, from the goalies on out.

Lately, the corollary to that has been their struggle to score goals.

“I know,” center Mathew Barzal said. “I mean, geez, we’re waiting for a four- or five-spot. But at this point in the season, it’s tough to score and we’re playing teams that know what they’re getting when they play us. So they’re trying to lock it down and do the same thing. At the end of the day, we’re getting chances. If we weren’t getting chances, that would be an issue.”

The Islanders scored one or zero goals for the third time in four games Thursday night in a 2-1, four-round shootout win over the Devils, who got a turn-back-the-clock performance from struggling goalie Cory Schneider at Prudential Center.

Thomas Greiss also was superb. He made 30 saves for the Metropolitan Division-leading Islanders (31-16-6) and stopped Kyle Palmieri, Nico Hischier, Drew Stafford and Jesper Bratt in the shootout.

The Isles swept the four-game season series from the Devils.

“All season, both goalies have given us a chance to win every game,” said Josh Bailey, who scored the lone goal in the shootout. “Going into the game, it already gives you an edge before the puck even drops having that confidence in your goaltending. Tonight, he was there when we needed him the most.”

The Islanders played one forward short after Cal Clutterbuck exited with a lower-body injury at 5:50 of the second period.

Schneider, who has not won in the NHL since Dec. 27, 2017 (he is 0-15-4 in his last 19 decisions), made 27 saves for the last-place Devils and denied Jordan Eberle, Barzal and Brock Nelson in the shootout before Bailey converted. Schneider made his first NHL appearance since Dec. 14 after recovering from an abdominal strain.

“He was definitely playing like the old Cory Schneider,” said Barzal, who grew up in Coquitlam, British Columbia, and used to watch Schneider play for the Canucks. “I’m happy we won, but it’s nice to see him get back on track a little bit.”

The Islanders went 0-for-4 with six shots with the man advantage on Thursday and are 1-for-17 in their last five games, including 1-for-5 in Tuesday night’s 3-1 loss at Boston. Then again, the Islanders killed off all three of the Devils’ power plays.

“We want to score a little bit more,” Islanders coach Barry Trotz said. “We’ve had some great goaltending battles. Everybody ramps it up to another level.”

Defenseman Ryan Pulock sprung Barzal, coming out of the penalty box, and he beat Schneider with a breakaway backhander to tie the score at 1-1 at 13:42 of the first period. Kevin Rooney, finishing off an odd-man rush, had beaten Greiss over his glove from the left circle at 5:32 of the period.

Trotz elevated rookie Michael Dal Colle to Barzal’s line with Bailey while struggling Anthony Beauvillier was dropped to Valtteri Filppula’s third line with Leo Komarov. Dal Colle also took Clutterbuck’s spot on the second power-play unit after he was injured.

Schneider looked sharp from the start with an early right-pad save on defenseman Johnny Boychuk in the first period and a subsequent stop on Barzal at the crease. In the second period, Schneider smothered Anders Lee’s power-play chance at the crease. At 10:30 of the third, Schneider reached back with his glove to stop defenseman Scott Mayfield’s shot from the point. He also robbed Beauvillier late in regulation and Barzal at the buzzer.

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