Kostopoulos put his own rebound past Martin Biron shortly after a faceoff in Philadelphia's zone for his third goal of the playoffs.
"It was exciting," Kostopoulos said. "It was a weird game for us. We were expecting such intensity with both teams coming off the big Game 7s and I think it was a hard game for everyone to get into, for the crowd to get into.
"Lucky enough (Saku) Koivu got that nice faceoff and Kovalev put it in to bail us out and we got the lucky bounce in overtime."
Kovalev drew Montreal even at 3 with 28.6 seconds left in the third, scoring a power-play goal off a faceoff with goalie Carey Price pulled for an extra attacker.
"The puck was in the scrum there and then all of a sudden it just came out," said Kovalev, who also scored a short-handed goal in the second to tie it at 2. "It was just perfect timing."
Flyers center Jeff Carter's stick broke as he lost the faceoff to Koivu, and Kovalev beat Biron with a shot into the top right corner.
"It was a tough break but there's really not much you can do when you break your stick," Carter said. "I don't even know where the puck went after that - obviously Kovalev got it, but there's not really much you can do."
The Flyers continue to have problems protecting leads. Philadelphia also blew two-goal leads in Games 1 and 6 in of its first-round series against Washington, and lost both games.
"We've just got to keep going," said Flyers forward Jim Dowd, who scored in the first to put Philadelphia up 2-0. "We're a good skating team. They've got great defensemen over there and we've got to make them go back as much as we can. Our strength is skating, getting the puck in deep and forechecking. We've got to make them go back every single shift, that's the key for us."
Joffrey Lupul scored a power-play goal 19 seconds into the third to give the Flyers a 3-2 lead.
Lupul, who scored Philadelphia's series-winning goal in overtime Tuesday night in Washington, got his second of the playoffs - after a brief video review - when the puck came out of Price's glove and hit Lupul's hand before going into the net off the Flyers forward's skate.
Canadiens coach Guy Carbonneau kept Price on the bench for a sixth attacker as Montreal began a power play in the Flyers' zone with 1:09 remaining after Philadelphia's Mike Richards was sent off for kneeing after he collided with Kovalev.
The strategy ultimately paid off as Kovalev scored his second goal of the game - his fourth of the playoffs - to send the game to overtime.
"(Kovalev) was well-rested because he didn't practice the last two days," Carbonneau said.
R.J. Umberger also scored in the first for Philadelphia.
Andrei Kostitsyn and Kovalev tied it at 2 when they scored in the second for Montreal, which won all four meetings between the two teams during the regular season.
Price stopped 30 shots in the first playoff meeting between the teams in 19 years, the fifth Montreal-Philadelphia series overall.
Biron also made 30 saves.
Kostitsyn, who failed to score on a penalty shot 6:32 into the second, made the most of a second chance when he took a pass from his younger brother, Sergei, and put a shot over Biron's right arm for his fourth goal of the playoffs at 9:44.
Kovalev tied it with a short-handed goal at 16:03 that necessitated a lengthy video review.
Referee Don Van Massenhoven ultimately pointed to the faceoff circle to the delight of the sellout crowd of 21,273 after replays supported the on-ice call that Kovalev's stick had made contact with the puck below the level of the crossbar off a rebound of Tomas Plekanec's shot.
"When the puck went in the air and I saw (Biron) was looking in the other direction, I knew I had enough time to let the puck come down a little bit lower to make sure it wasn't a high stick," Kovalev said.
Biron raced from his crease to argue with Van Massenhoven near the blue line, to no avail.
The game had more than its share of odd goals, beginning with Philadelphia's first at 13:15.
Umberger got credit for his second goal of the playoffs when Montreal defenseman Patrice Brisebois deflected the Flyers forward's centering pass intended for Patrick Thoresen into the right side of the net.
Dowd, who played for Montreal late in the 2003-04 season, made it a 2-0 lead with his first goal of the playoffs when he drove a one-timer from the slot past Price at 16:49.
Notes: Flyers D Jaroslav Modry was not in the lineup for a third straight game and will return to the Czech Republic on Saturday to attend the burial of his father, who died last week after a seven-month battle with colon cancer. ... Philadelphia and Montreal met in the playoffs twice in the 1970s, and two more times in the 1980s. The Canadiens won a semifinal in 1973 and swept the Flyers in the 1976 finals. Philadelphia won a conference final against Montreal in 1987 and lost another in 1989. ... Dowd has eight career playoff goals.
