Andrej Meszaros and Scott Hartnell also scored for Philadelphia, which had been slumping. The Flyers had lost five of eight, but moved into fifth place in the Eastern Conference with the win.
Josh Bailey scored two goals and Kyle Okposo had one for the Islanders, who have the fourth-lowest point total in the NHL.
Evgeni Nabokov stopped just 24 of 29 shots for New York. Nabokov was sensational in a 1-0 shootout win over the Flyers last month. He turned aside 47 shots, including two in the shootout. But the veteran goalie clearly wasn't the same this time against a team that had been shut out in two of its three previous games.
Ilya Bryzgalov made 21 saves to earn the win in his fifth straight start.
Jagr and Read scored 39 seconds apart in the second period to put Philadelphia ahead 4-2.
First, it was the 40-year-old Jagr making a move reminiscent of his glory days in Pittsburgh. Jagr carried the puck down the right side, skated around the net, fended off a defender and backhanded a shot past Nabokov for his 663rd NHL goal.
It was Jagr's 1,642nd career point, moving him ahead of Joe Sakic into eighth place on the NHL career list. Former teammate Mario Lemieux is ahead of Jagr at 1,723 points.
Read then gave the Flyers a two-goal cushion by streaking down the center and deflecting a perfect crossing pass from Wayne Simmonds in for his 18th goal. Read is tied with Colorado's Gabriel Landeskog for the most goals among rookies.
Hartnell scored his 31st goal off a rebound to make it 5-2 in the third period. Sean Couturier then added an empty-net goal.
Bailey's second goal of the game tied it at 2-2 in the opening minute of the second period. A centering pass from Frans Nielsen set up Bailey's shot from the far side of the net.
The Flyers went up 2-1 in the last minute of the first period when Meszaros scored a power-play goal off a pass from Read. Meszaros ripped a slap shot off the post, but the puck bounced in off Nabokov's back.
Meszaros and Read teamed up to tie it 1-1 midway through the first. Read deflected Meszaros' slap shot past Nabokov.
Bailey gave the Islanders a 1-0 lead just 1:35 in. He took Casey Cizikas' pass from behind the net, made a nifty move and beat Bryzgalov.
NOTES: The Flyers lost James van Riemsdyk indefinitely after the forward broke a bone in his left foot blocking a shot in the first period. ... The Islanders lost for the first time in three games in Philadelphia this season. They were 0-11-2 over the past four seasons in Philadelphia before winning the first two meetings this season. ... The Flyers lead the NHL with 54 power-play goals. ... Read hadn't scored since Feb. 2. ... Hall of Famer Gordie Howe watched the game along with his son, Mark Howe, who was recently elected into the Hall. ... Jagr's was his 60th against the Islanders. He has more points against the Islanders (143) than any other team.