James Daniel, the nation's leading scorer who had 38 points, hit a bank shot with six seconds left for Howard (6-5), which was without three injured starters but got Daniel back after he missed a loss to Purdue.
Davis, who finished with 23 points, missed a shot at the end of regulation and Howard missed two shots as the first overtime ended.
Radford (6-5) led by nine at halftime.
Radford freshman Ed Polite Jr. had his best game with 27 points and 11 rebounds.
Marcel Boyd had 17 rebounds with 15 points for Howard.
Radford coach Mike Jones, a former Howard player and assistant, returned to his alma mater for the first time as a head coach.