The Dragons held a 32-20 advantage 35 seconds into the second half before Rice (1-5) responded with a 17-3 spree to move ahead for the first time, 37-35, with 11 minutes, 42 seconds to play.
Former Frankford High star Keith Washington, a freshman, made a scoop shot to put the Owls ahead.
Massenat scored five points during an ensuing 7-0 spurt that gave Drexel a 42-37 lead. Then Rice drew within 44-42 with 7:35 left - and did not score for the next 7:20.
"We actually played some team defense," Flint said. "We started switching, making their guys drive and making them make plays at the rim."
Dartaye Ruffin's play proved pivotal for the Dragons. After rebounding Max Guercy's missed jumper that would have tied the score, Ruffin ended the possession with a four-foot jumper that extended the Dragons' lead to 46-42 with 6:44 to play.
Ruffin terminated Rice's next possession with another rebound, then got fouled while shooting and made both free throws to give Drexel a 48-42 lead with 5:43 left. The 6-foot-8 junior again foiled the Owls by making a diving steal.
"I thought Dartaye did a great job of switching off and playing guards," Flint said, "and not letting them have straight line drives but making them take tough shots."
Tamir Jackson led Rice with 20 points. Julian DeBose added 12 points and nine rebounds. Washington finished with two points, an assist and three turnovers in 18 minutes.