Goins scored 23 points on 9-of-14 shooting with 11 rebounds, while Moore had 22 points on 9-of-15 shooting for the Dukes (4-6).
The Spartans (2-6), playing their first game in over two weeks, were led by Trevis Simpson's 24 points.
A.J. Davis's layup sparked a 13-1 James Madison run about 6 minutes in for a 21-9 lead.
UNC Greensboro cut James Madison's 49-29 halftime lead down to 64-61 with 7:05 left, but Goins hit two baskets and the Spartans never got within five points again.
James Madison is 5-0 all-time against UNC Greensboro as the teams met for the first time since 1999.
The game was a homecoming for UNC Greensboro coach Wes Miller, who played his freshman year at James Madison in 2002-03 before transferring to North Carolina.