Jordan Ferguson stepped in front of a Paul Mroz pass and returned it 34 yards for a score with 2:04 left in the game to seal the win for Brown (1-2).
Rhode Island (0-5) led 24-20 when Mroz hit Phadrae White from 13 yards out with 5:12 left in the third quarter.
Seth Rosenbauer answered when he scored from the 4. The Rams retook the lead 31-27 when Harold Cooper returned the ensuing kickoff 92 yards for the score. The Bears then marched 91 yards in seven plays and grabbed the lead for good when Fuller hit Alexander Jette for a score with 5:16 remaining.