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.
Kapita was the top-rated recruit from West Virginia in the class of 2015 and recently played at DME Academy in Florida.
With the Suns drafting Tyler Ulis, they now have five guards on their roster who attended Kentucky.
NBA fans will be looking forward to their newest players making plays like this next season.