The Bruins' defense did more than its share, as well, intercepting six of Utah quarterback Travis Wilson's passes, including on the last drive after Wilson led the Utes to the UCLA 23 with less than a minute left.
Hundley, who went 17 for 27 for 211 yards, also had an interception, throwing an ill-advised pass into the flat that Keith McGill picked off and returned 19 yards for a touchdown to tie it at 24 with 13:23 left.
But the sophomore quarterback followed with a meticulous, 13-play, 48-yard drive - completing a pair of passes to convert third downs, then scrambling for 8 yards to set up the go-ahead field goal.
Shortly after Anthony Jefferson's second interception of the night, Hundley ran 36 yards to put the Bruins (4-0, 1-0 Pac-12) ahead 34-24 and they held on after Utah made a field goal to pull within seven, then recovered the onside kick.
Moments after the game ended, a snow squall began over Rice-Eccles Stadium, where UCLA has had no luck in the recent past. This snapped a two-game losing streak at Utah that spanned six years and two coaches. More importantly, the Bruins established themselves as the early team to beat in a Pac-12 South division in which every other team has at least one loss and their cross-city rivals, Southern California, has two.
Meanwhile, Hundley made a case as one of the most versatile quarterbacks in the country.
Combined with his 85 rushing yards, Hundley finished 4 yards short of reaching the 300-yard mark in total offense for the fourth time in four games this season.
And fitting for a guy who can do it all, he even had a 27-yard quick kick, which pinned the Utes down on their 8 in the first quarter.
This game between two fast-break offenses started fast, as expected, with both teams scoring touchdowns over the first five minutes.
It settled into more of a tactical game from there and Hundley made better choices, completing passes to nine receivers and even catching a pass himself.
That score tied it at 14 and set up a quarterback-versus-quarterback showdown that Wilson lost badly.
But Utah's sophomore quarterback couldn't be blamed for all those picks. In fact, the first four came off tipped balls, a couple of which hit his receivers in the hands. He finished 22 of 44 for 288 yards and two scores, including a 54-yard connection to Dres Anderson for Utah's first touchdown.
But Wilson's six picks led to 17 UCLA points - a number that couldn't be ignored in a game decided by a touchdown.
Paul Perkins led the Bruins with 92 yards rushing on 16 carries.