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.
Former Michigan point guard Spike Albrecht will play for Purdue as a graduate transfer next season.
Only six major-conference teams have failed to win an NCAA tournament game since the turn of the century. When will each program snap the streak?
Bruce Weber's Kansas State team earned a share of the Big 12 title three years ago, but a lot has changed since then and the pressure is on.