Leading 94-93, Chad Posthumus grabbed an offensive rebound off a miss by Turner and was fouled by Elijah Pittman. Posthumus converted two free throws to start a 10-1 run by the Eagles to close out the game.
Kareem Canty's 3-pointer with a minute left in the first half gave Marshall a 21-point lead, its biggest of the game, 48-27. Morehead State (3-1) closed the half with a 6-0 spurt to head into the break trailing 48-33.
The Eagles held 16-8 advantage on the offensive glass and outscored Marshall 21-3 in second-chance points.
Posthumus scored 14 points and pulled down 15 rebounds, seven offensive.
Elijah Pittman tied a career-high with 31 points for Marshall (2-1). Kareem Canty added 19 and nine assists but had nine turnovers.