Voracek scored on a wrist shot from the slot to deny St. Louis its first sweep of a homestand of five or more games in its history.
"I think one guy changed, one guy got caught," Voracek said. "Usually when you have a bad change in overtime, it's on the rush. I thought it was going to be 2-on-1, then I had to slow down a little bit to have a little bit of time and space. I had nothing other than that, so obviously it worked out."
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Brian Elliott made 30 saves, and Sean Couturier had two assists for the Flyers (25-16-6), who blew a 3-1 lead in the third period but won for the third time in four games.
"It was certainly a really solid game on our part," Flyers coach Alain Vigneault said. "We've played against some real tough opponents lately, Washington, Tampa Bay, and Boston, and to come in here tonight against the Stanley Cup champions, play the way we did, hard at both ends, there were some performers on the ice that were making it real challenging for us to keep our focus and our cool, but we did and we battled and we found a way to grind out and get a greasy road win against a real tough team."