Kopitar scored twice, and Copley made 28 saves for the Kings (27-17-6), who have won their past two games after losing three in a row for the second time this season.
Los Angeles coach Todd McLellan said Fiala's line with Blake Lizotte and Jaret Anderson-Dolan was supposed to be out on the ice at the end of the third period, but he kept them on the bench so Fiala would be ready for overtime.
"I don't think he was real happy with that, but that was us thinking ahead," McLellan said of Fiala, who had his 15th multipoint game. "Thank goodness we had him out there fresh and he was able to finish."
James van Riemsdyk had a goal and an assist, Ivan Provorov had two assists, and Carter Hart made 34 saves for the Flyers (20-21-8), who have at least one point in 10 of 14 games (9-4-1).
"Tonight, I mean, I watched how quick that team played when the game started and I said, 'Man.' But then I thought we stuck right in there with them, got our legs and played a really good hockey game," Philadelphia coach John Tortorella said.
Samuel Fagemo tied it 3-3 for Los Angeles at 18:36 of the second period on a wrist shot from the slot through traffic.