Cole's 3-pointer gave the Hawks an 85-81 lead. Jordan Reed brought the Bearcats back within two on a floater but Evan Cooper and Yolonzo Moore II both made free throws down the stretch to ensure the victory.
Yosef Yacob's 3-pointer for Binghamton tied the game at the end of regulation.
The victory was the fifth in six games for the Hawks (15-15, 9-6 America East), who are tied with Albany for third place.
Mark Nwakamma led five Hawks in double figures with 19 points. He also grabbed 12 rebounds for his sixth double-double this season.
Reed led the Bearcats (7-21, 4-11) with a career-high 33 points - the most by any America East player in a conference game this season. He added 13 rebounds for a conference-best ninth double-double.