The fourth-quarter score gave Memphis a 38-28 lead, but Rice closed to 38-35 with 2:09 left on Chase Clement's 20-yard scoring pass to James Casey.
Rice failed to recover an onside kick, and Memphis (3-4, 2-1 Conference USA) held on for the win.
Joseph Doss carried the ball 26 times for 179 yards and a touchdown for the Tigers. Duke Calhoun had four catches for 119 yards.
Jarrett Dillard scored three touchdowns for the Owls (1-6, 1-2), who outscored Memphis 21-18 in the second half but could not stop Doss and Hankins, who completed 21 of his 32 passes for 306 yards and three touchdowns.
Clement, who was intercepted twice, completed 28 of his 43 passes for 289 yards. He threw for four touchdowns and ran for another. Dillard caught three touchdowns from Clement, one in each quarter, and finished with five catches for a team-high 85 yards.
The Tigers had little trouble moving the ball in the first half and took a 20-14 lead into halftime when Matt Reagan made the second of his two first-half field goals with less than 10 seconds left.