The Knights (5-2, 3-0 Conference USA) are off to their best start since 1998, setting up next Saturday's game against East Carolina for the lead in the league's East Division.
The Owls (2-6, 1-3) hardly threatened, falling behind 34-0 before their first score midway through the third period.
Weaver ran for touchdowns of 1, 4 and 2 yards. He carried 15 times for 53 yards.
Godfrey, who didn't play in the fourth quarter, controlled the game. He completed 13 of 18 passes for 178 yards and a touchdown. He also ran 10 times for 41 yards and another score.
UCF led 27-0 by halftime, dominating from the opening kickoff. The Knights struck quickly, using a 59-yard, play-action pass completion from Godfrey to Kamar Aiken on the first play from scrimmage to set up their opening score.
Godfrey scored on a 5-yard touchdown run just 1:23 into the game. He passed for his second touchdown, a 3-yard strike to tight end Adam Nissley on a fourth-and-goal.
Weaver scored two touchdowns in the first half, from 1 yard in the first quarter and from 4 yards in the second. His third rushing touchdown, from 2 yards, was set up by a fumble when Rice quarterback Nick Fanuzzi was sacked by defensive end Troy Davis.
After trailing 34-0, Rice finally scored midway through the third quarter on a 16-yard pass from Fanuzzi to running back Sam McGuffie, who finished with 65 yards on 16 carries.
The Owls scored again with 6:33 remaining on a 1-yard run by Jeremy Eddington, closing the gap to 41-14. Rice free safety Travis Davis got his first career interception late in the third period, but the Owls were quickly forced to punt.
UCF showed little regard for the Rice defense, converting on all four of its fourth-down attempts, including a 40-yard touchdown pass from backup quarterback L.D. Crow to Ricky Kay with 9:07 remaining.
The Knights have won nine consecutive Conference USA Games, including their last six in 2009. They are off to their best start since the 1998 season, outscoring opponents 214-97 through their first seven games.