The Rams (7-4) led 37-21 with 1:21 remaining in the opening half before a three-point play from DeAndray Buckley cut the New Hampshire deficit to 13 at halftime.
Rhode Island led by as many as 14 and no fewer than eight during the second half and forced 17 turnovers, while turning it over only seven times.
Tyrone Conley led the Wildcats (4-4) with 16 points, Buckley scored 11 and Ferg Myrick 10.
The Rams came into the game shooting 45.3 percent from the floor as a team, but shot just 21 of 59 (35.6 percent) and 13 for 22 from the free-throw line (59.1 percent).
Nikola Malesevic scored 11 points for Rhode Island.
Myrick was ejected from the game at the 4:07 mark of the opening half after receiving two technical fouls following a scuffle with Ben Eaves.