The host Tigers (9-0) were led by Jabari Brown’s 22 points. Jordan Clarkson added 21 points to help Missouri extend its home winning streak to a nation-best 24 games.
UCLA (8-1) suffered its first defeat of the season. Kyle Anderson and Zach LaVine each scored 13 points, while Norman Powell scored 11 points to round out the Bruins' list of double-digit scorers.
Two free throws by Johnathan Williams III gave Missouri a 62-60 lead it would not relinquish with 9:30 remaining. The team trailed for most of the first half and into the second before Earnest Ross made a 3-pointer with 14:46 left.
Saturday’s win helped Missouri push its run of consecutive wins against non-conference opponents at Mizzou Arena to 79. The victory was the first in that stretch against a nationally ranked team.
Jordan Adams scored 22 points and had 10 rebounds for UCLA (8-1), which dropped its first road game this season. Kyle Anderson and Zach LaVine each added 13 points.
The Tigers remain the only unbeaten team in the Southeastern Conference and are off to their best start since the 2006-07 season. UCLA entered the game averaging a Pac-12 leading 90.6 points.
Missouri trailed 43-35 at halftime after committing 12 turnovers and shooting 41.7 percent from the field, but used back-to-back fastbreak dunks by Ross and Brown to pull within 49-45 with 16:44 left.
After UCLA's Tony Parker converted a free throw, Ross then hit consecutive 3-pointers to give the Tigers their first lead since 11:20 left in the first half.
Brown's 3-pointer five minutes later widened Missouri's lead to 67-62, and UCLA couldn't get any closer the rest of the way.
Ross finished with 20 points and Williams grabbed 15 rebounds as Missouri shot 38.5 percent from behind the arc after starting 1 of 7.
The Bruins shot 50 percent from the field and forced 12 Missouri turnovers in the first half, but shot 8-of-31 in the second half.
The Bruins' 71 points were a season-low, one point fewer than they scored in the opener against Drexel.
The teams traded baskets for the first 10 minutes before a 3-pointer by Bryce Alford sparked a 14-0 run that gave the Bruins a 30-17 lead with 6:47 left before the break. Missouri answered with the next 11 points, but UCLA then countered with the following eight for a 38-28 lead. LaVine finished the run with a thunderous windmill dunk.
Both schools played in the Las Vegas Invitational last week and were selected co-champions after defeating Nevada and Northwestern in Nevada. UCLA and Missouri agreed to not play each other ahead of Saturday's matchup.
The game ended a home-and-home series after UCLA defeated the Tigers 97-94 in overtime at Pauley Pavilion last Dec. 28.