Jamad Williams had 14 carries for 72 yards, including a 6-yard touchdown in the first quarter as the Golden Hurricane improved to 4-1 on the season and 2-0 in Conference USA.
Damaris Johnson caught seven passes for 85 yards as Tulsa tied its all-time series with Rice. Each team has won seven times and there was one tie.
Ryan Lewis completed just 12 of his 32 passes for 99 yards and a touchdown, and Charles Ross led the Owls (0-5, 0-2) with 33 yards on 13 carries.
Lewis found tight end Taylor Wardlow with a 29-yard pass in the back of the end zone to tie the game at 7-7 after Williams' early TD. Rice went ahead on a Clark Fangmeier field goal late in the second quarter, but before the half ended Kevin Fitzpatrick kicked a field goal that allowed Tulsa to tie the game at 10-10.
Kinne put Tulsa ahead for good midway through the third quarter with a 6-yard touchdown run.
The Golden Hurricane defense has not allowed a rushing TD this season.