Nabokov was sensational, turning aside every shot he faced — including two in the shootout.
The last-place Islanders are 6-1-2 in their last nine games and 11-5-2 in their past 18.
Coming off losses to New Jersey and the New York Rangers last weekend, the Flyers hoped to rebound against a team they used to dominate. But they couldn't get one past Nabokov.
Nielsen scored on the first shot in the shootout. After Nabokov stopped Danny Briere, Tavares fired one past Bryzgalov. Nabokov then stuffed Wayne Simmonds to end it.
Nabokov recorded his 52nd career shutout and second this season in his return to the NHL after a brief stint in Russia. He registered his 305th win, tying former Islanders great Billy Smith for 22nd on the all-time list.
The Flyers were 6-0 against New York last year and 2-1 this season. The Islanders snapped a 13-game winless streak in Philadelphia with a 4-1 victory last month.
Nabokov, runner-up for the Vezina Trophy in 1998, played part of last season in Russia's KHL and declined to report to the Islanders after being claimed off waivers. But he returned to the NHL this season and has played well in the absence of No. 1 goalie Rick DiPietro. Nabokov entered with a 2.31 goals-against average and .920 save percentage. He improved to 12-12.
Nabokov stopped a pair of Scott Hartnell's one-timers in the first two periods and stoned Claude Giroux on a rebound in front late in the period. He also made a sprawling save on Max Talbot's short-handed breakaway shot in the second and blocked Jaromir Jagr's point-blank slap shot a few minutes later.
He made a couple more big saves on Philadelphia's power play in the opening minute of the third, including a pad stop on Braydon Coburn's slap shot from the point.
After being released by SKA St. Petersburg last year, Nabokov wanted to play in North America. He signed a one-year contract with the Detroit Red Wings in January. But NHL rules forced him to go through the waiver process before going to Detroit and he was claimed by the Islanders.
Nabokov refused to report, choosing to stay home on the West Coast rather than play for a non-contending team.
Bryzgalov's best stop was a kick save on P.A. Parenteau in the first period. He stuffed Nino Niederreiter's backhander right after the Flyers killed off a penalty in the second.
Notes: G Sergei Bobrovsky didn't start for Philadelphia despite a 7-1 career record against the Islanders. ... Briere returned after missing six games with a concussion, but James van Riemsdyk sat out for the 11th straight game with his concussion. ... D Milan Jurcina (elbow) and D Travis Hamonic (broken nose) didn't play for New York. ... The Flyers are 120-85-31 against the Islanders. ... New York D Aaron Ness made his NHL debut. ... Tavares has 11 goals and 15 assists in the last 18 games. ... Jagr has scored more points against the Islanders (142) than any other team.