The Wildcats (6-13, 2-3 Mid-Eastern Athletic) finished 24 of 27 from the foul line, including 15 of 16 in the second half with the miss coming with 19 seconds left.
Johnson was 9 of 9 from the line and Brandon Stewart hit 6 of 6 while scoring 11 points.
Zech Smith scored 13 points and grabbed 14 rebounds for the Bears (4-16, 2-4), who were 11 of 20 from the foul line. Cedric Blossom and Blake Bozeman had 12 points apiece.
Johnson had 14 points as Bethune-Cookman took a 31-26 lead at halftime. Morgan State got within one with six minutes left but the Wildcats held them off and then made seven free throws in the final 1:37.