Will Triggs added 20 points, Anthony Campbell had 14, and Thomas Greer chipped in with 11 for Austin Peay (5-12, 1-2).
After Triggs' free throw put the Governors ahead 72-70 with just 11 seconds remaining in the second period, Drew Kelly converted a layup for Morehead State to tie it with two seconds left to send the game into overtime.
With Morehead State (8-9, 2-1) trailing 81-79 with 1:13 to go in the overtime period, Taariq Muhammad's layup with 28 seconds left tied the game at 81.
Betran answered with a 3-pointer with just one second left on the clock to give the Governors the win.
Bakari Turner led the Eagles with 19 points.