Brodeur, pulled after allowing four goals in two periods of a 4-0 road loss to the Islanders on Monday, lasted the entire game in the rematch and picked up his NHL-leading 31st win in his 29th consecutive start.
"It's nice to get a few bounces offensively," Brodeur said. "The guys worked extremely hard, especially in the third. We created a lot of chances and didn't give up much at all."
The Islanders mustered just three shots in the third period and turned over the puck nine times. New York lost for the second time in seven games.
"We didn't push the pace to allow ourselves the opportunity to win the game," Islanders coach Scott Gordon said. "We played the game not to lose."
Salvador's left-point shot off a faceoff beat screened goalie Rick DiPietro as the Devils won for just the second time in six games. Rob Niedermayer won the draw cleanly to set up the defenseman's first goal since Dec. 18.
"It came through a screen and snuck through me," DiPietro said. "It's kind of how the third period went. You hope you don't look back and say those two points will cost you the playoffs."
Travis Zajac assisted on the Devils' first two goals and Parise scored an empty-netter with 45.1 seconds left. New Jersey entered with just eight goals in 12 games before cracking the four-goal plateau for the first time in nine games.
"It was nice to go into the third tied after giving up two power-play goals," Parise said. "For us to play a solid game and feel good about it was the most important thing. It was a huge win for us."
After Langenbrunner opened the scoring, New York turned Colin White's major boarding penalty into power-play goals by Okposo and Hillen almost two minutes apart in the second period.
White drove Sean Bergenheim hard into the penalty box-side boards, resulting in a game misconduct at 9:15. The bloodied Bergenheim skated to the dressing room and then returned in the final period.
Okposo jump-started the Islanders' offense 19 seconds after White's penalty, scoring the tying goal by beating Brodeur low glove side from just outside the right circle.
Hillen's high-slot slap shot 2:02 later put the Islanders ahead 2-1. The defenseman netted his first goal since Nov. 7 after his shot hit the post before crashing off Salvador and into the net.
Brodeur gave Salvador a friendly pat after the goal. The defenseman made up for the bad bounce with his second game-winning goal this season.
"I told him not to worry about it," Brodeur said. "It wasn't a mistake, it was reaction. In a situation like that, it does no good to start yelling."
New Jersey was held without a shot in the second period before Parise scored with 32 seconds left. The rare power-play strike for the Devils tied the game despite the Islanders owning a 14-1 edge in shots during the period.
Parise, who added his team-leading 23rd goal in the last minute, marked the Devils' first power play goal in nine games and ended an 0 for 23 drought.
"It was a relief because we've been working hard on the power play to get goals," Devils coach Jacques Lemaire said. "The guys were really happy we got it. You could feel it on the bench. Everyone was excited even though it was a lucky goal."
The Devils rebounded from a home loss on Friday to Montreal and nearly lost momentum after New York's power play woke up.
"It was a tough kill. I thought we dominated play before that," Brodeur said. "We took over the game after that and didn't let ourselves down because we had a hiccup in the middle of the game."
The last time he faced the Devils on Monday, DiPietro recorded his first shutout in nearly two years. But Langenbrunner fired in a low-right circle one-timer to end a repeat bid 5:52 into the game.
Zajac helped set up the goal with a crisp behind-the-net pass to the open Devils captain, who scored his first goal in 10 games.
"We didn't match their intensity in the first period," Okposo said. "In the third period, we just didn't have it. That was the game right there."
After generating five first-period shots, the Islanders allowed the Devils to dictate the pace in the third.
"We didn't get our forecheck established and we weren't putting as much pressure on their [defense] in the offense zone," DiPietro said. "We got away from it in the third and need to learn from our mistakes."
Devils winger Jay Pandolfo appeared in his 800th NHL game, all with the organization that drafted him in 1993. ... Langenbrunner hadn't scored since posting a hat trick against Minnesota on Jan. 2. ... New York banged three shots off the post before scoring against Brodeur. ... Hillen ended a 34-game goal drought.