Bradley shot a two-under 68, while Dufner posted a one-under 69 on Sunday at Atlanta Athletic Club.
Bradley overcame a triple-bogey at the 15th hole with a birdie at 16 and a long birdie putt at 17.
Dufner, after Bradley's miscue, was five ahead, but also hit in the water at 15. He made bogey, then bogeyed 16 and 17 as well, before two-putting for par at the last to join Bradley in the playoff.
Neither golfer has won a major championship.
Bradley is a tour rookie playing in his first major, but he won the Byron Nelson Championship this year.
Dufner, who tied for fifth at the PGA last year, is winless on tour.
They will play 16, 17 and 18 and the man with the lowest score wins.
Denmark's Anders Hansen missed the playoff by a stroke after his final-round 66 left him at minus-seven.