American Bradley had led the tournament for the first three rounds and began the final morning in Texas leading Bae by one shot, but that advantage quickly disappeared on the front nine.
The South Korean carded four birdies on the first eight 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 theten 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 his first US PGA Tour victory with a two-putt par.
The 26 year old from Daegu said: "The course was really tough.
"The winds were very strong. I did very well this week.
"I told myself, 'hang in there and just keep trying, keep trying, focus on my game'. That's it. That's what I did."
He continued: "I felt my swing was very comfortable this week. I hit a bunch of good shots and I made a lot of putts.
"This is my first win and I'm very happy, so excited. I was a little nervous on 16, 17, 18, but I tried to focus on my game and I won so I'm very happy."