With the victory, the Islanders moved two points ahead of the idle New York Rangers into the eighth and final Eastern Conference playoff spot.
The Islanders (18-16-3) have 11 games remaining while the Rangers have 13. The Islanders and seventh-place Devils each have 39 points but New Jersey has a game in hand.
The Islanders haven't reached the postseason since 2007.
"We played with pace and sustained our defense-first mentality," Islanders coach Jack Capuano said. "We talked last summer about how we wanted to play meaningful games the last month of the season. Well, now we have them."
Nielsen's power-play goal at 19:32 of the second period came when a rebound of a Mark Streit slap shot ricocheted to him in the right circle. Nielsen fired the puck past Ondrej Pavelec for this fourth of the season.
"It came right to me and thankfully it went in," Nielsen said. "We battled hard, got a lead and played our game."
Matt Moulson added his 12th of the season at the 18:00 mark of the third. And John Tavares scored his team-leading 23rd of the season into an empty net with 36 seconds left.
The Jets (18-18-2) lost their fourth straight, but still lead the Southeast Division by two points over the Washington Capitals and four over the Carolina Hurricanes. Winnipeg lost Monday night to the Rangers at Madison Square Garden.
"We didn't play well either night," Pavelec said. "We have to figure out something soon."
Poulin was making his third start of the season to give a rest to Evgeni Nabokov, who has started all but six of the team's 37 games. The 22-year-old Poulin has been the backup to Nabokov since his recall from AHL Bridgeport on Feb. 23.
His best save was a sprawling stop on an Alex Burmistrov shot in the closing seconds.
"I felt good out there and we worked hard from start to finish," Poulin said. "It was a satisfying win for sure."
Islanders center Anders Lee opened the scoring in his NHL debut on his first shot on goal at 16:31 of the first period when his wrist shot eluded a screened Pavelec.
"I just wanted to get it on net," said Lee, who was given the milestone puck by linemate Casey Cizikas. "Obviously, I am happy with the result."
The 22-year-old Lee signed with the Islanders on Monday after completing a stellar college career at Notre Dame where he was team captain.
A sixth-round pick by the Islanders in 2009, the 6-foot-3, 220-pound Lee led the Central Collegiate Hockey Association in goals (20) and points (38) this season.
Postma, a seventh-round pick by the Jets' former incarnation -- the Atlanta Thrashers -- in 2007, scored his second of the game and third of the season at 6:02 of the second.
Colin McDonald tied the score at 2 at 9:56 of the second when he tapped in a rebound for his seventh of the season.
The Islanders, who are 11-5-1 on the road, won for just the seventh time in 20 home games (7-11-2). Seven of their remaining 11 games are on the road.
The Islanders are 17-2-2 when scoring at least three goals and 1-14-1 when scoring two or fewer.
"We're seeing the results of doing all the little things right," Capuano said. "The guys showed great energy all game. We have to keep winning one-on-one battles to be successful."
The victory was the 78th as coach of the Islanders for Capuano, moving him past Peter Laviolette into third place on the franchise wins list behind Al Arbour (740) and Terry Simpson (81).
This was the second meeting between the teams. The Jets won 5-4 in overtime at home on Jan. 27. They meet again April 20 in Winnipeg.
The Jets scratched defenseman Grant Clitsome, who was hit near his eye with an errant stick the previous night against the Rangers. ... The Islanders placed defenseman Joe Finley on injured reserve to make room for Lee on the roster.
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UNIONDALE, NY - APRIL 02: Nik Antropov #80 of the Winnipeg Jets looks on during warm-ups prior to the game...