The Hornets (13-12, 10-3 Mid-Eastern Athletic Conference) trailed 68-66 with 3:27 remaining before Tate hit a 3 and a jumper off an offensive rebound to spark an 8-0 run the rest of the way.
Tate finished with five offensive and five defensive rebounds.
Marques Oliver had 20 points and 10 rebounds for the Hornets, including a free throw to send it to OT.
Oliver made the first of his two foul shots with 28 seconds left in regulation to tie it, 64-64. He missed the second, which was brought down by Tate. Casey Walker missed a 3-pointer on that possession, which Oliver rebounded but his shot was blocked with 1 second left.
Wesley Dunning had 19 points and 13 rebounds for Hampton (9-18, 5-8).