Kopecky, who had been a scratch the previous five playoff games and was in the lineup because of an injury to Andrew Ladd, scored from the left side with a sharp-angled shot that beat backup goalie Brian Boucher at 8:25 of the third.
Game 2 is Monday night at the United Center.
Danny Briere and Scott Hartnell had a goal and two assists apiece for the Flyers.
Chicago's Troy Brouwer scored two goals, the second putting the Blackhawks ahead 5-4 in the second period and prompting the Flyers to replace starter Michael Leighton with Boucher. Ville Leino, Blair Betts and Arron Asham also scored the Flyers.
Dave Bolland had a short-handed breakaway goal for the Blackhawks, and Patrick Sharp and Kris Versteeg also scored for Chicago.
Brouwer's second goal, on a pass from Marian Hossa who reversed himself behind the net, came from the left circle with 4:42 left in the second, sending ex-Blackhawks goalie Leighton to the bench.
Leighton had been brilliant since taking over in the second round. He entered the game with a 6-1 record, including three shutouts, after replacing an injured Boucher in the Eastern Conference semifinal series against the Boston Bruins. Leighton entered with a 1.45 goals-against average and save percentage of .948. He was pulled after giving up his fifth goal in just 20 shots.
Chicago's Antti Niemi made 27 saves for the win.