Running back Phillip Lindsay had 11 receptions, 149 total yards and a touchdown for the Buffaloes (7-2, 5-1), which sits atop the South Division.
Colorado won its third straight despite being called for 12 penalties for 128 yards, including eight personal fouls, and committing four turnovers.
UCLA had 13 penalties for 96 yards and missed three field goals, including one that was blocked.
Liufau finished 19-for-30 for 143 yards passing and ran for 58 more despite missing the last 5:43 of the first half after he was sacked by UCLA defensive lineman Takkarist McKinley.
Liufau fumbled and was knocked out of the game for two series. Backup Steve Montez came on and threw an interception.
Liufau returned at the start of the second half and led Colorado to a pair of field goals.
Mike Fafaul was 15-for-29 for 185 yards passing and one touchdown and Darren Andrews had six catches for 88 yards and a score for UCLA (3-6, 1-5).
The Buffaloes trailed 10-7 at halftime but Oliver’s 42-yard punt return led to Chris Graham’s 21-yard tying field goal.
Colorado took a 13-10 lead on another Graham field goal, this one from 44 yards out early in the fourth quarter.
Oliver sealed it with his punt return with 5:27 left.
Colorado went ahead 7-0 on Lindsay’s 1-yard plunge with 5:49 left in the first. Linebacker Kenneth Olugdobe’s interception and return into UCLA territory set up the score.
The Bruins wasted no time in getting even. After a long run back on the ensuing kickoff and the Buffaloes’ third personal foul gave UCLA the ball in Colorado territory, Fafaul hit a wide-open Andrews for a 39-yard touchdown with 4:04 left in the first.
Liufau’s second interception of the first half gave UCLA the ball at Colorado’s 28, but the Buffalo defense stiffened. The Bruins lost three yards and settled for JJ Molson’s 48-yard field goal to go up 10-7.
UCLA got the ball back when defensive lineman Takkarist McKinley’s sack of Liufau caused a fumble, and Jayon Brown returned it Colorado’s 18.
Colorado stiffened again and Chidobe Awuzie blocked Morton’s 26-yard attempt to keep it a three-point game.
NOTES: Colorado DL Jimmie Gilbert was ejected for targeting early in the first quarter. … UCLA is 12-7 record in the month of November since 2012. … Sefo Liufau’s interception in the first quarter was his first in 15 pass attempts, the longest stretch without throwing one in Colorado program history. … The Bruins’ defense ranks seventh nationally in pass-efficiency defense.