Devils goaltender Cory Schneider made 28 saves.
"I think we wanted to get a lead against a team like this, especially a team that can shut it down," Strome said. "We played a solid game and needed to get off this losing skid. We knew it would be a greasy game and we battled pretty hard."
Strome's goal 7:37 into the first period was his second game-winner of the season.
"(Halak) make a couple of big saves on the 2-on-1 [in the third period] and made a couple of glove saves so he's been great for us all year and we needed him to be," Strome said. "I think he's got that mojo back for the playoffs. He'll be huge for us down the stretch."
Halak, who missed two games because of a lower-body injury, earned his first shutout since a 27-save, 3-0 win against the New York Rangers on Jan. 13. He won all four games of the season series against the Devils. The Islanders denied the Devils on their only power-play opportunity in the second period and are 22-for-22 on the penalty kill spanning nine games.
"(Halak) gave us a chance to win again and had some unbelievable stops and some puck luck because a few could have gone either way, but he gave us a chance and that's why he's a great goaltender," Islanders defenseman Johnny Boychuk said. "He was probably the main reason we won."
Halak's best sequence of saves came midway through the third period when he denied Steve Bernier and Scott Gomez off a rebound at 7:46. Devils co-coach Lou Lamoriello pulled Schneider for an extra attacker with 2:39 remaining, but the Devils could not get one past Halak. Okposo scored an empty-net goal with 37.4 seconds left.
"Once they got that second [goal], it was a tough climb for us but I thought we matched the intensity and matched their effort," Schneider said. "It was a good atmosphere, a fun game. The crowd was into it and the guys were into it, but we came out on the wrong end.
"Their goaltender was a little bit better than I was and it's probably not a good thing for our team when that's the case."
Hamonic connected on a one-timer from the left circle off a pass from Strome in the third period that beat Schneider high to give the Islanders a 2-0 lead at 12:19. Hamonic's fifth goal of the season came after Stephen Gionta's penalty expired.
Halak made 11 saves in the second period, including a pad save against Adam Henrique off a slap shot from the right circle at 9:22. He gloved a point-blank slap shot from the right circle by defenseman Eric Gelinas at 11:11 and later got his right toe on a backhand attempt from the slot by Henrique at 12:49 before capping a busy second with a pad save off a redirect from the slot by Gomez at 16:22.
"It's always hard to play against teams desperate for points and trying to make it to the playoffs; I've been in that situation," Halak said. "You really have nothing to lose because you just want to get those points. For us, we need to play that kind of desperate hockey the rest of the season. We need to keep getting points. Look at the standings, it's getting tighter and tighter."
The Islanders, who begin a four-game homestand against the Minnesota Wild on Tuesday, are five points behind the New York Rangers for first place in the Metropolitan Division with nine games remaining. The Rangers have three games in hand.
"The game had a similar feel to the playoffs; there wasn't a lot of time and space and you had to fight through sticks and checks and interference but that's part of the game this time of the year," Islanders captain John Tavares said. "We came out with a little more intensity and desperation and played a solid game."
Tavares earned in his first point in five games when he assisted on Okposo's empty-net goal.
Strome gave the Islanders a 1-0 lead 7:37 into the game when he took a pass from Nikolay Kulemin in the left circle and took a wrist shot that beat Schneider. The Devils goalie played the puck into the left-wing corner but Gelinas was rubbed out of the play by Kulemin and fed Strome entering the circle. The goal was the first for Strome in nine games and his 15th of the season.
"It was a great individual effort [by Kulemin] and it's always nice to be rewarded," Strome said. "He's a workhorse with the puck. I was calling for it and his bread-and-butter is on the forecheck, so he kind of gets pucks back and it was a great play by him."
Schneider kept the Devils within a goal with a blocker save from the slot off a rebound by Kulemin 5:56 into the second. The save came three minutes after Devils center Travis Zajac rang the long-side post from the left circle.
"You don't want to put too much pressure on yourself this time of the year," Halak said. "You just keep stopping pucks; I got a little lucky when they hit the post but these points are huge. We need to keep improving until the end of the year."
Islanders defenseman Nick Leddy returned to the lineup and had 21:18 of ice time. He missed four games because of an upper-body injury.
The Devils (31-30-11), who had their three-game winning streak end, will play the Los Angeles Kings on Monday. Forward Patrik Elias returned to the lineup after missing the past three games with back spasms. Left wing Mike Cammalleri missed his second straight game due to the flu and Michael Ryder was a healthy scratch.
Follow Mike Morreale on Twitter: @MikeMorrealeNHL