The Governors (10-6, 7-1 Ohio Valley Conference) won their seventh straight as Murray State (7-10, 5-4) blew a 17-point first-half lead.
Austin Peay led 62-59 with 3:05 left before reserve Landon Shipley hit four free throws and Reed hit a layup to finish with 17 points and seven rebounds.
Fernandez Lockett had 16 points and a team-high eight rebounds for Austin Peay and Shipley finished with 13 points, including 8-of-8 free throws.
The Racers had their largest margin at 40-23 with 2:48 left in the first half on two free throws by Marvin Williams.
The Racers missed all nine of their 3-pointers in the second half, ending up with 4-of-17. Austin Peay hit 5-of-18 for the game.