SIU-Edwardsville (3-12, 0-3), which has lost five in a row, went scoreless over the final three-plus minutes of the first half and trailed 42-29 at the break.
Yemi Makanjoula's layup pulled the Cougars within one with five minutes left, but Bradds scored four during a 9-2 run that put the Bruins up 78-70.
Keenan Simmons made a 3 to cut SIU-Edwardsville's deficit to five with 1:26 remaining, but Bradshaw hit a 3-pointer 29 seconds later and Belmont held on from there.
Simmons led the Panthers with a career-high 21 points on 10-of-12 shooting. C.J. Carr had a career-best nine assists.