After Henry made 1 of 2 tries, Ohio’s D.J. Cooper, whose layup with 35 seconds left tied the game, missed a long-range shot at the buzzer.
The Eagles (4-4), who had lost their four previous games against Division I opponents this season, trailed 25-20 at halftime but opened the second with a 9-0 run, capped by two jumpers by Henry.
They led 35-28 with 14:11 remaining after consecutive 3-pointers by Andre Smith, but the Bobcats (7-3) scored the next five points and stayed within five the rest of the way.
Smith and Joab Jerome added 10 points each and Larry Brown nine for Winthrop.
Reggie Keely scored 14 points for Ohio, making all six of his first-half shots but taking none in the second half.