The Mountaineers (7-8, 3-2) led throughout the second half, taking a nine-point lead, 48-39, on Omar Carter's 3-pointer with 8:08 remaining.
The Paladins (7-9, 1-4) cut the margin to two points and trailed 52-50 on Brandon Sebirumbi's jumper with 2:35 left.
After two unsuccessful possessions by each team, Trice's dunk made it 54-50 with 36 seconds to play. Carter later added two free throws.
Furman led 20-10 with 11:03 left in the first half after Charlie Reddick's 3-pointer, but Appalachian went on a 15-3 run and took the lead on Neal's 3-pointer 4:31 before halftime. Neal scored the Mountaineers' final eight first-half points.
Sebirumbi led Furman with 18 points and Reddick had 14.