Johnathan Franklin rushed for 79 yards and a TD for the Bruins (5-2, 2-2 Pac-12), who bounced back solidly from a blowout loss to California that knocked them out of the Top 25 last weekend.
Shaq Evans caught a 64-yard TD pass from Hundley, who showed off his uncommon freshman poise while winning a duel with Travis Wilson. Utah's freshman quarterback passed for 220 yards in his first career start, but the Utes (2-4, 0-3) lost their third straight.
UCLA's defense shut out the Utah offense until Wilson led a 90-yard drive capped by Jon Hays' 1-yard TD pass to Dres Anderson with 3:16 to play.
Anderson, the son of former UCLA receiver Flipper Anderson, had nine catches for 70 yards for the Utes, who have never won at the Rose Bowl. Utah is off to another winless start in Pac-12 play after starting 0-4 last year before a strong finish.
Utah got its only points of the first three quarters when Ryan Lacy recovered a muffed punt return by UCLA's Steven Manfro in the end zone late in the first quarter.
Hundley went 15 for 21 in another impressive performance against a strong defense, showing off his smarts and athleticism while leading several lengthy drives. Franklin moved closer to the top spot on UCLA's career rushing list with another grind-it-out effort, most notably picking up a crucial first down with just over 2 minutes to play.
After an unimpressive effort by Utah's offense last week against Southern California, coach Kyle Whittingham shook up his lineup by giving the start to Wilson, a product of Orange County's San Clemente High School who had played in every game this season as a backup. Hays had started the Utes' past three games since they lost Jordan Wynn to a career-ending shoulder injury.
But Andrew Abbott intercepted a tipped pass by Wilson on the freshman's opening drive, and UCLA mounted a 13-play scoring march capped by Hundley's expert ball fake on a 12-yard TD run up the middle.
Utah evened it 65 seconds later thanks to Manfro, who inexplicably attempted to field a booming punt while backpedaling furiously toward the UCLA goal line. Manfro missed two chances to control the ball before Lacy picked it up in the end zone.
Hundley put the Bruins back in front 67 seconds later with a sideline pass to Evans, who ran untouched into the end zone after a Utah defender gambled on an interception.
After UCLA opened the second half by stopping Utah on fourth down at the Bruins 34, Hundley calmly engineered a 14-play scoring drive capped by Franklin's 3-yard TD run.
Utah finally got moving in the waning minutes, reaching the UCLA 3 with a trick play featuring receiver DeVonte Christopher throwing a 23-yard pass to Wilson.
Wilson then rushed inside the UCLA 1 on third down, but his helmet came off on the play, forcing Hays to take his first snap. The senior calmly hit Anderson, who stretched over the goal line for Utah's first offensive TD.
Utah didn't try an onside kick, and UCLA picked up a first down while running the clock down to 8 seconds before punting. The Utes' final trick play went nowhere.