It was the fifth career hat trick for the Flyers center. He tipped in a Kimmo Timonen shot during a power play midway through the third period to forge a 2-2 tie for Philadelphia.
Briere also scored in the initial frame and fought Ottawa's Kyle Turris while sparking the Flyers to a win in the first of a weekend home-and-home series with the Senators.
Ilya Bryzgalov turned aside 35-of-37 shots and made a key save on Bobby Butler in overtime to earn the victory.
Butler and Milan Michalek found the net for Ottawa, which is 7-2-1 in its last 10 games.
Anderson finished with 36 saves.