On the game-winning play, Gustav Nyquist skated in on Ducks goaltender Jonas Hiller on a breakaway. The netminder sprawled to make a poke-check, but the puck came right to Brunner, who tapped it in to end the game.
Brendan Smith and Pavel Datsyuk each netted a third-period goal for Detroit, which notched its second overtime victory in the Western Conference quarterfinal series.
Jimmy Howard turned aside 31-of-32 shots to backstop the Red Wings.
Dave Steckel and Matt Beleskey found the net for second-seeded Anaheim, which who seek to regain the advantage at home in Wednesday's Game 5.
Hiller finished with 46 saves in the hard-luck loss.