Chris Horton hit two free throws with 21 seconds left to give Austin Peay a 52-49 lead. Cornell Johnston missed a 3-pointer on Eastern Illinois' next possession and Davis made 1 of 2 free throws with 7 seconds left for a four-point lead and the Governors held on from there.
Austin Peay trailed by 13 points early in the second half before going on a 13-0 run, capped by a Robinson 3-pointer, to knot the game at 36.
Eastern Illinois (12-8, 6-2) has lost back-to-back home games after winning nine straight overall.
Trae Anderson led the Panthers with 22 points. Johnston added 11 points and had eight assists.