The 21-year-old Swedish goalie kept his team in the Eastern Conference playoff race with a 1-0 win over the New Jersey Devils on Sunday.
He also became the first goalie in Islanders history to record a shutout for his first win.
"This is what your dream about, so it feels amazing," said Nilsson, a third-round draft pick by the Islanders in 2009. "But most importantly, we got the two points we needed."
Kyle Okposo scored the game's only goal for the Islanders (28-29-9), who moved within five points of Winnipeg for the eighth and final Eastern Conference playoff spot.
New York recalled the 6-foot-5 Nilsson from AHL Bridgeport on Feb. 29 after he posted a 6-0-1 mark in February and was selected the league's goalie of the month.
"I felt more confidence in my game as we went along," Nilsson said. "Now I know what to expect as I move forward in my NHL career."
Nilsson was making just his second career start against the 39-year-old Brodeur, who was playing his 1,177th game for New Jersey, which fell to 36-24-5 (77 points).
"You have to throw more pucks at a young guy," Martin Brodeur said. "And we have to play harder because it will bite you in this league if you don't. We just didn't create much offense in front of that young goalie in the pipes."
The teams played a scoreless first period before Okposo scored his 15th goal of the season at 2:39 of the second, beating Brodeur to the glove side with a low wrist shot from the slot.
The goal was assisted by rookie center Casey Cizikas, his third assist in six games since his recall from AHL Bridgeport.
"I'm so happy for Anders since I've played with him all season in Bridgeport," Cizikas said. "Today we put together a very good team effort. We have to keep it going."
Nilsson made a key save with 1:54 remaining on a wrist shot by Devils right wing Petr Sykora, who was playing his 1,000th game.
The 6-foot-5 Swedish goaltender made his first NHL start on Nov. 21, losing 5-0 at Pittsburgh. He also played the last two periods in relief of Rick DiPietro in a 6-0 home loss against Boston on Nov. 19.
Islanders coach Jack Capuano, who stresses defense, was impressed with the poise of his young goaltender and his franchise mark.
"Anders had a great camp, played extremely well for Bridgeport and now he has confidence here," Capuano said. "He was aggressive, played his angles well and made big saves when he had to. That gave us a chance to win."
The Islanders and Devils were both playing for the third time in less than 72 hours and each coming off wins. New Jersey beat the Capitals 5-0 in Washington on Friday while the Islanders ended a four-game trip with a 3-2 win in Boston on Saturday.
New York lost 6-3 at Philadelphia on March 1 and 3-2 in overtime at Washington on Feb. 28.
The teams hadn't met since splitting two games on Thanksgiving weekend. The Devils won 1-0 at the Nassau Coliseum and the Islanders won 3-2 in Newark.
Sunday's game was the first of three meetings between the teams in a seven-day stretch. The Islanders play three straight against New Jersey, while the Devils host the New York Rangers on Tuesday before playing the Islanders again on Thursday and Saturday.
"Obviously, we have to get to the net more," Devils right wing David Clarkson said. "Every game means everything with how close the standings are. We can't sit here and dwell. We have to get ready for next game."
The Islanders failed to capitalize on a two-man advantage for 1:08 midway through the second period. They have yet to score a goal with a 5-on-3 advantage this season.
Capuano credited his club's work ethic for the weekend's back-to-back one-goal victories.
"All of our guys gave good minutes," Capuano said. "We've battled and played hard these past two games and it has paid off."
- The Devils were without injured left wing Alexei Ponikarovsky (right knee) and defenseman Andy Greene (back spasms).
- The Islanders scratched forward Michael Grabner because of the flu.
- The Devils are tied with Los Angeles and Montreal for the most goals scored (six) while playing 5-on-3.
- The Devils have scored and allowed 13 short-handed goals, most in the NHL in both categories.