Munford scored 14 points in the second half, including 11 during a 16-3 run when the Rams (5-8) extended a five-point lead to 43-25 with 4:20 remaining.
Both teams shot poorly in the first half, which ended with the Rams ahead 21-14 despite shooting 25.8 percent (8 of 31). The Bears (4-8) shot 25 percent (5 of 20). Rhode Island's Andre Malone scored 10 of his 12 points in the half.
Powell's 3-pointer put Rhode Island ahead to stay, 12-10, with 10:27 left until halftime.
Rhode Island has won 10 of its last 11 meetings with Brown and leads the series 101-53.
ESPN's Dan Dakich joins The Paul Finebaum Show to discuss the SEC's presence in the NBA Draft.
Ted 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.