American Bradley had led the tournament for the first three rounds and began the final round in Texas leading Bae by one, but that lead quickly disappeared on the front nine.
The South Korean carded four birdies on the first seven holes while Bradley bogeyed the par-four third, finding the water in the process, to give Bae a four-stroke lead.
But a double-bogey on the ninth and bogeys on the 10th and 15th, on which Bradley picked up his first birdie of the day, saw the pair tied for the lead once again.
Bae, however, moved back in front on the 16th when Bradley, who won the tournament in 2011, circled the outer rim with a putt from four feet.
Bradley could not recover to save par on the 17th and Bae was two shots clear on the final hole, with a fine approach shot allowing him to seal victory with a two-putt par.
The 26-year-old from Daegu, who shot a final-round one-under 69 to finish at 13 under, told pgatour.com: "The course was really tough.
"The winds were very strong. I did very well this week."