Bradley parred the final two holes to complete a bogey-free round of 6-under 64, ending at 13-under-par 267.
That proved to be sufficient when Furyk -- who had led the entire round -- stumbled to a double-bogey on the 18th hole, finishing at minus-12.
He missed a 5-footer for bogey that would have forced a playoff, but settled instead for a share of second place with a 1-under 69.
Steve Stricker also fired a 64 to claim a piece of the runner-up spot on the South Course at Firestone Country Club.
Claiming sole possession of fourth place was Louis Oosthuizen (1-under 69) at 11-under 269.