The win was the American's first major success of his career.
Bradley began the day one shot behind Dufner, who gradually pulled away from his fellow countryman.
However, Dufner squandered the five shot lead he had amassed to allow his opponent back in the contest, and both players finished on eight-under.
Bradley, 25, held his nerve during the play-off to claim the victory.
Anders Hansen finished one shot back on seven-under, while Robert Karlsson, David Toms and Scott Verplank were tied on five-under.