Damien Thigpen also ran for a touchdown for the Bruins (6-2, 3-2 Pac-12), who rebounded from road defeats at Stanford and Oregon despite a sluggish start in their homecoming game.
Fuller caught a 76-yard TD pass for the Bruins' first score and added a rushing TD with 3:36 to play. Hundley passed for 273 yards and scored on runs of 11 and 1 yards while moving into third place on UCLA's career total yardage list.
Sefo Liufau passed for 247 yards for the Buffaloes (3-5, 0-5), who have lost 13 consecutive Pac-12 games.
Colorado hasn't beaten a ranked team on the road in 21 tries since 2002.
The Bruins won despite 11 penalties for 122 yards, steadily pulling away from Colorado with solid defense and just enough playmaking from Hundley, their sophomore quarterback. Hundley finished with 345 total yards, passing Tom Ramsey and Cory Paus on UCLA's career yardage list in just his 22nd game with the Bruins.
Christian Powell rushed for 97 yards, Tony Jones rushed for a score, and Will Oliver kicked three field goals for the Buffs. Paul Richardson shook off an injury to catch eight passes for 81 yards and a touchdown.
Back home, UCLA's defense got some help from the offense against Colorado. The Bruins have outscored their opponents 199-66 in four victories at the Rose Bowl this season.
After opening the game with two fruitless drives, UCLA's offense had a yardage-losing play to open its third drive. On the next play, Hundley delivered his long strike down the middle to Fuller, who caught the ball in heavy coverage and kept his feet for a score - the longest pass in the careers of both Hundley and Fuller.
Colorado went back ahead in the second quarter with a 76-yard drive ending in Liufau's TD pass to Richardson, but the Bruins responded with another strong drive capped by Hundley's 11-yard TD run up the middle.
UCLA's Cameron Judge then forced a fumble on Colorado's ensuing kickoff return, and Hundley hit Fuller for a 6-yard TD catch capped by a pirouette at the goal line for an 11-point lead.
Colorado trailed 21-13 at halftime despite possessing the ball for more than 20 minutes and outgaining the Bruins, who committed 75 yards of penalties in the first half.
Hundley and Thigpen scored in the third quarter, and Fuller added his short TD run with 3:36 to play.
Colorado began the afternoon with a promising victory, winning the opening coin toss for the first time in eight games this season.