The Flyers built a 4-1 lead before Chicago scored twice in a span of less than 4 minutes of the third period.
Philadelphia's lead was cut to 4-3 with 4:10 to play but Leighton fought off an almost constant Chicago attack and the Flyers clinched it on Jeff Fisher's empty net goal with 25 seconds remaining.
Having tied the NHL's championship series, the Flyers will try to take the lead Sunday when Chicago hosts Game 5. Philadelphia lost the first two games of the series and was trailing in the third period of Game 3 before coming back to win it in overtime.
Mike Richards, Matt Carle and Claude Giroux all scored in the first period to give Philadelphia a 3-1 lead and the Flyers held onto that advantage throughout the second.
Ville Leino then put a wrist shot past Antti Niemi 6:43 into the third and the Flyers appeared headed for an easy victory. Chicago made its comeback on goals by Dave Bolland and Brian Campbell. The score by Campbell that brought the Blackhawks within one was his first of the playoffs.
Niemi stopped 26-of-30 shots.
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