Brett Hundley, a redshirt freshman, was 21 of 28 for 202 yards and two touchdowns and rushed for 68 yards and a touchdown. Joseph Fauria caught three passes for 53 yards and a touchdown.
Franklin just missed his career high of 216 yards rushing set in 2010 against Washington State. He became the first player in UCLA history to have two rushes of at least 72 yards in a game after scoring on touchdown runs of 74 and 78 yards in the first half.
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Rice's Taylor McHargue threw for 172 yards and two touchdowns on 17 of 28 passing and rushed for 95 yards and a touchdown on 22 carries. Sam McGuffie caught eight passes for 63 yards and a touchdown.
UCLA finished with 651 total yards in Jim Mora Jr.'s first game as coach, compared to 347 for Rice.
After a scoreless third quarter, UCLA put the game away with a 3-yard touchdown pass from Hundley to Fauria to make it 41-24 after the two-point conversion failed with 12:29 left in the game.
Franklin extended the lead to 49-24 with a 22-yard touchdown run and Hundley connected with David Allen for the two-point conversion with 7:54 remaining in the fourth.
Hundley took UCLA's first snap and ran 72 yards down the right sideline for a touchdown. The PAT was blocked to make it 6-0.
After Sheldon Price's interception of McHargue, UCLA took three plays to go 47 yards and score on an 11-yard pass from Hundley to Jerry Johnson. The PAT was blocked again to make it 12-0. Franklin took UCLA's first snap on its next drive 74 yards to extend the Bruins' lead to 19-0.
The Owls got on the board with a six-play, 75-yard drive capped by an 8-yard pass from McHargue to Luke Wilson to cut the lead to 19-7 with 6:08 remaining in the first and sliced the lead to nine on a Chris Boswell 53-yard field goal four minutes later.
After Rice recovered a Bruins' fumbled punt return, the Owls trimmed the lead to 19-17 on a McHargue 1-yard touchdown run to begin the second quarter.
UCLA rebounded to extend the lead to 22-17 on a 27-yard field goal by Ka'imi Fairbaim with 9:17 remaining in the first half, and Franklin went 78 yards on the second play of UCLA's next drive to up the UCLA lead to 29-17 3:36 before halftime.
The Owls answered with a 2-yard touchdown pass from McHargue to McGuffie 1:20 before the half to make it 29-24, but Damien Holmes returned Turner Peterson's fumble 43 yards for a touchdown 17 seconds before the half to extend the Bruins' lead to 35-24 after another blocked PAT.
Rice's Cameron Nwosu set an NCAA record with three blocked PATs in one game.