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.
The Blackhawks are in their first final series since 1992 and are aiming for the franchise's first championship since 1961. The Flyers, who last made the finals in 1997, are shooting for their first title since the Broad Street Bullies won the second of two straight championships in 1975.
The Flyers weren't too bullish Saturday night. They played the entire game without a penalty.
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.
The Flyers scored a late goal in each of the first two periods. Asham converted from the left circle after a cross-ice pass from Briere with 1:11 remaining in the second, tying it at 5-5.
Fourth-liner Betts picked up a loose puck after it came off the boards when the Blackhawks couldn't control it, skated into the left circle and fired it past Niemi for his first playoff goal, putting the Flyers back in front 4-3 in the second.
But a little more than two minutes later, Versteeg got his own rebound and tied it after Kopecky had skated behind the net. The Flyers capitalized in the closing seconds of the first when Briere fired a shot from the slot, hustled toward the net and scored on the rebound for his 10th goal of the playoffs, stunning the Blackhawks with just 27 seconds left in the period for a 3-2 lead.
But former Flyers forward Sharp quickly got the Blackhawks even.
On a break with Hossa on the other side of the ice, Sharp held onto the puck and fired from the right circle, beating Leighton just 71 seconds into the second and tying the game.
The Flyers, behind four shots from Briere, had a 17-9 advantage in the opening period, showing that they have the quickness offensively to match the Blackhawks.
Bolland took the puck away from defenseman Braydon Coburn at center ice, started his breakaway, and his shot struck Leighton before finding its way into the net. Bolland's second short-handed goal of the playoffs gave the Blackhawks a 2-1 lead.
But after Chicago's third penalty of the opening period, the Flyers converted on the power play. Flyers defenseman Chris Pronger drove a shot from the high slot, and Hartnell scored on the rebound to make it 2-2.
On Philadelphia's first goal, Leino directed the puck toward the crease from the left circle. Niemi blocked it and then it apparently deflected off Chicago's Niklas Hjalmarsson and went into the net just 6:38 into the game to give the Flyers the early lead.
The raucous crowd of 22,312 at the United Center wasn't quieted for long.
Just 68 seconds later, Hossa made a nice backhanded pass from the right circle to Brouwer in the circle. From high in the slot, he rifled it past Leighton to tie the game.
