Bae ended with a 13-under-par 267 at the TPC Four Seasons Resort, corralling his first career PGA Tour victory.
The South Korean-born player benefited when 2011 champion Bradley encountered trouble down the stretch and ended with a round of 2-over 72, two shots off the pace at minus-11.
Taking sole possession of third place was Charl Schwartzel (68) at 10- under-par 270.
Fourth place fell to Justin Bolli, who fired a 5-under 65 Sunday to finish at minus-9.
Knotted in fifth place one shot further back were Martin Kaymer (68), Scott Piercy (72) and Morgan Hoffmann (66).
Swim Week Miami Beach 2014 [PHOTOS]