In the final round at TPC Four Seasons Resort, Bae shot a 1-under 69 for a four-round total of 13-under 267.
Bae, 27, is the third South Korean to win on the PGA Tour, following Choi Kyoung-ju, 43, and Yang Yong-eun, 41.
He was awarded prize money of US$1.17 million (1.3 billion won).
After three rounds, Bradley was the leader with 13-under, but the winner of the 2011 PGA Championship shot a 2-over 72 in the final round.
Charl Schwartzel of South Africa placed third with 10-under.
reporter
Park Sae-jin
swatchsjp@ajunews.com