The Buffaloes (2-0) trailed 14-10 with five minutes remaining before marching 63 yards in 11 plays for the winning score.
After freshman Mason Crosby's extra-point attempt was blocked by Brandon Chillar, UCLA (0-1) had a chance to win if it could move into position for a field goal.
But after Maurice Drew fumbled the ensuing kickoff and returned it only to the 10-yard line, backup quarterback Drew Olson, who replaced injured starter Matt Moore late in the first quarter, threw incomplete on four straight passes against a suddenly stout Colorado defense.
A roughing-the-passer penalty against the Bruins triggered Colorado's winning drive. Klatt completed an 8-yard pass to Derek McCoy on third-and-7 and a 10-yarder to D.J. Hackett on third-and-3. Rolling right, Klatt, who earlier left the game for one play after being hit in the head, found Klopfenstein in the right side of the end zone.
Colorado's win spoiled the head coaching debut of UCLA's Karl Dorrell, who was an assistant at Colorado for six years in the 1990s.
It was one of several Colorado connections for the Bruins, who hired assistant coaches Jon Embree and Eric Bieniemy and strength and conditioning coach Doc Kreis from the Colorado staff in the offseason.
Colorado mounted two long scoring drives in taking a 10-7 halftime lead.
UCLA enjoyed favorable field position throughout the first half, including midway through the first period when Chris Kluwe's 46-yard punt was downed at the Colorado 2-yard line by Keith Short.
But Colorado responded with a 98-yard, 11-play touchdown drive, capped by Bobby Purify's 1-yard run with 2:21 left in the period.
Purify started the drive with an 18-yard carry, Jeremy Bloom went 19 yards on an end-around and Klatt passed 14 yards to D.J. Hackett to set up the score.
Two plays later, UCLA lost Moore when he was hit low on a pass play and injured his left knee.
Early in the second quarter, UCLA's Nnamdi Ohaeri downed a punt at the Colorado 3. When Colorado couldn't make a first down and punted, UCLA had possession at the Buffaloes' 42.
Midway through the quarter, Colorado fashioned a 57-yard, 15-play drive culminating in Crosby's first career field goal, a 40-yarder.
Early in the third quarter, UCLA marched to the Colorado 12, including Olson's 16-yard pass to Marcedes Lewis on third-and-10, but Justin Medlock's 35-yard field goal attempt was wide left.
The Bruins came right back for a go-ahead score after Colorado tailback Brian Calhoun's fumble at the Colorado 37. The drive appeared to have stalled on holding and offensive pass interference penalties. But, facing a third-and-30, Olson completed a 38-yard pass to Lewis to the Colorado 8.
After a false start, Olson hit Lewis for 13 yards on a slant pattern for a touchdown and a 14-10 lead with eight seconds left in the period.
INJURY UPDATE ON HARRIS:
Senior DE Marques Harris underwent surgery for about 90 minutes late Saturday evening for a broken tibia and fibula in his left leg, which occurred late in CU's 16-14 win over UCLA earlier in the day.
The surgery, which commenced around midnight, was performed by team orthopedic surgeon, Dr. Eric McCarty, and it went well.
Harris is expected to be released from Boulder Community Hospital on Monday and will start rehab later in the week. He is out for the season, but does have a redshirt year available to him. He is on schedule to graduate next May.
No other players made or were added to the injury report for the game.