The Bears ended their season with a four-game winning streak that began with a stunning 52-24 decision over previously unbeaten Kansas State.
Baylor raced to a 35-10 halftime lead with quarterback Nick Florence leading the Bears on touchdown drives of 53, 79, 75, 60 and 56 yards. Florence threw two touchdown passes and Martin ran for a score in each of the first three quarters.
Lache Seastrunk rushed for 138 yards and produced a 43-yard Baylor touchdown. Martin finished with 100 yards on 21 carries.
UCLA's Brett Hundley was 25-of-49 for 295 yards and two touchdowns without an interception.
Baylor closed out the season with an 8-5 record and UCLA was 9-5.
The Bears were the first of eight teams from the Big 12 Conference that will see action in bowl games this season. The Pacific 12 Conference slipped to 1-2 in this year's bowl games with four teams yet to play.