Goins scored six points during a 12-0 second-half run that gave the Dukes (2-5) a 54-34 lead with 9 minutes remaining.
The Eagles (2-4) never got closer than eight points after that.
Winthrop led 21-13 in the first half after a 3-pointer by Derrick Henry. But James Madison responded with a 15-3 run to lead 28-24 with 5:04 left until halftime. Semenov scored five points, and Goins and Charles Cooke each made three-point plays during the Dukes' rally.
Andre Smith led the way for Winthrop, scoring 26 points and making 4 of 6 from 3-point range.