The 26-year-old South Korean closed with a 1-under 69 to finish at 13-under 267.
Bradley was trying to become the Nelson's first wire-to-wire winner since Tom Watson in 1980. He set the TPC Four Seasons course record with an opening 60, but finished with a 72 on a day with wind gusting to near 40 mph at times.
Bae already had 11 international victories — on the Korea, Japan and Asian tours.
After four birdies in an early five-hole stretch, Bae had a four-stroke lead. Bradley got even at No. 15, but missed a short birdie putt on the next hole.
