Twyman and Langston each scored five points during a 12-0 second-half run in which Quinnipiac came back from a 42-38 deficit to take a 50-42 lead with 12:35 remaining.
The Rams (6-4) got back within three points on four occasions, the last on a jumper by Will Martell that made it 69-66 at the 2:01 mark. They never scored again, missing three 3-point attempts, two free throws and turning the ball over twice.
Twyman and Johnson added two free throws apiece in the final 11 seconds to secure the win for the Bobcats.
Delroy James, who came in averaging 21.4 points, led Rhode Island with 18.