Taylor Bessick nearly had a double-double with nine points and 13 rebounds for the Dukes (9-8, 3-1 Colonial Athletic Association), who won their second straight game.
The score was tied at 38-all with 4:43 left in the second half after Dartaye Ruffins' 3-pointer. But James Madison pulled away with a 10-0 run to claim a 48-38 advantage, holding Drexel scoreless for four minutes.
Moore completed a three-point play and made two free throws during the James Madison spurt.
Frantz Massenat scored 11 points for the Dragons (5-11, 1-3), who lost their third consecutive game.
James Madison shot 37 percent (17 of 46) from the floor, while Drexel made 36.2 percent (17 of 47) of its shots.
Maryland point guard Melo Trimble has withdrawn from the NBA draft and will return to College Park for his junior season, sources told ESPN.
Louisville's Chinanu Onuaku, who recently underwent a medical procedure to correct a heart rhythm issue detected during the combine, will keep his name in the NBA draft pool.
Andrew White III will return to the Nebraska men's basketball team next season after removing his name from the NBA draft.