Marcus Neal's 3-pointer put Delaware State ahead 54-53 with 5:10 to play, but Maine (2-1) went on a 6-2 run from that point to seal the comeback win.
McNally's layup with 1:12 left gave Maine its final margin, and the Black Bears defense held the Hornets (2-1) scoreless the rest of the way.
Barnies finished with 10 points and 12 rebounds for the Black Bears, who won despite trailing 29-19 at the half. Junior Bernal also had 10 points.
Maine outscored Delaware State 40-27 in the second half.