Badau Diagne tallied 12 for Coastal Carolina (13-12, 8-6 Big South Conference). Raffa, who made 8 of 16 shots and four 3-pointers, gave the Chanticleers a 63-62 lead with 35 seconds left on his lone free throws of the night.
UNC Asheville (15-13, 9-5) regained the lead on D.J. Cunningham's dunk with 21 seconds showing, but Coastal Carolina called a timeout to set up Greenwood's game-winner.
With a chance to win it, the Bulldogs' Trent Meyer missed a 3-pointer at the buzzer.
Both teams shot 46 percent, 25 of 54 for Coastal Carolina and 23 of 50 for UNC Asheville, but the Chanticleers scored 34 points off 22 Bulldogs turnovers.
Jeremy Atkinson had 18 points for UNC Asheville, which was outscored 40-26 in the paint.