After going 3-0 down to the Penguins in the first period, Danny Briere started Philadelphia's comeback with a goal in the second and another midway through the third before Brayden Schenn tied the score at 3-3 to force overtime.
Jakub Voracek then scored the game-winning goal early in the extra period to give the Flyers the road victory in the opener of their best-of-seven series.
Familiarity has bred contempt between the Penguins and Flyers, who played each other three times in the final three weeks of the regular season, including an April 1 encounter studded with fights.
The home team came out strongly on Wednesday and took a 3-0 lead on goals from Sidney Crosby, Tyler Kennedy and Pascal Dupuis.
Briere, playing in his first game since taking a big hit against the Penguins that sparked the April 1 fights, got the visitors within striking distance then drew a penalty to put the Flyers on the power play.
Schenn took full advantage of the extra man to score the Flyers' third and send the game into overtime.
(Writing by Jahmal Corner in Los Angeles; Editing by Peter Rutherford)