The shootout game-winner salvaged a victory for the second-seeded Capitals, who squandered a 3-0 lead but still managed to pull to within a single point of Philadelphia atop the East.
Briere completed the Flyers' rally with a goal late in the third period that gave Philadelphia a 4-3 advantage but Marcus Johansson knotted the game with less than 4 minutes left in regulation.
Michal Neuvirth stopped 29-of-33 shots and Mike Knuble finished with a goal and two assists for the Caps.
Claude Giroux and Andreas Nodl each notched a goal and assist for the Flyers.
Sergei Bobrovsky started in net but was pulled in the first period after yielding three goals on nine shots and Boucher turned aside 13-of-14 shots the rest of the way.