The Islanders used a dominant third period to erase a 3-1 deficit and force overtime.
Brock Nelson made it a one-goal game at 8:16, and Okposo tied it with 2:58 left in regulation on a lunging push with his stick after the puck struck his skate. The goal was confirmed by video replay.
Frans Nielsen and John Tavares also scored in the four-round shootout against Ray Emery, who made 40 saves through overtime.
Anders Nilsson turned aside Vincent Lecavalier and Sean Couturier in the tiebreaker to help the Islanders avoid a four-game, season-series sweep versus the Flyers.
Tavares also scored a power-play goal in regulation for the Islanders.
Philadelphia is 15-2-3 against New York since the start of the 2010-11 season.
Matt Read did most of the offensive damage for the Flyers, netting two of Philadelphia's three goals in the second period, and joining Claude Giroux with goals in the shootout.
New York outshot Philadelphia 34-15 after the first period, including 17-3 in the third.
Read gave the Flyers a 1-0 lead with a short-handed goal and then stretched Philadelphia's advantage to 3-1 late in the second. Wayne Simmonds scored in between for the Flyers, who defeated New York with a come-from-behind 6-4 victory at home on Saturday.
Read, who had an empty-netter in that win, has scored six of his 13 goals this season versus the Islanders. He has two two-goal games this season against New York and has 10 goals in 14 career games with the Islanders.
Nilsson made 26 saves through overtime as he subbed for Kevin Poulin, who gave up five goals on 31 shots two days earlier at Philadelphia. Nilsson improved to 1-2-2 in his sixth outing of the season and 10th his NHL career.
Emery was sharp until late. He beat the Islanders in relief of Steve Mason on Saturday, stopping all five shots he faced in 26 minutes.
Read got the scoring going 1:26 into the second while teammate Nicklas Grossmann served a hooking penalty. Read raced with his puck all the way from his zone, moved around defenseman Andrew MacDonald in front and scored with a backhander.
Tavares got the Islanders even at 9:26 when he jammed in a return pass from Okposo at the right post for his team-leading 23rd goal.
Simmonds put the Flyers back in front, scoring off his own rebound at 11:30, and Read made it 3-1 with 1:44 left in the second. Philadelphia scored three times on just nine second-period shots.
Nelson got the Islanders to 3-2 when he scored shortly after an Islanders power play.
NOTES: The Islanders, who returned from a 4-2 trip, are 18-5-2 when they score at least three goals this season. ... Read played in his fourth game after missing six because of concussion symptoms. ... Poulin has won five of seven since taking over as New York's No. 1 goalie after Evgeni Nabokov was injured on Jan. 6 - the Islanders' previous home game. ... New York D Travis Hamonic missed his fourth straight game since sustaining an upper-body injury on Jan. 12 at Dallas.