Muhammad's 3-pointer gave UCLA a 95-93 lead and Earnest Ross was unable to tie the game for Missouri when he could hit just 1-of-2 free throws with 46 seconds left.
Travis Wear hit a jumper as the shot clock was winding down to boost the Bruins' advantage to three points.
UCLA forced the overtime when Jordan Adams hit a layup with 12 seconds to go in regulation that brought about an 88-88 tie.
The Bruins improved to 10-3 by winning their fifth straight game. The Tigers (11-2) had a six-game winning streak snapped.
Muhammad led the way with 27 points while Wear had 22. The Bruins shot 50 percent from the field, but could make only 4-of-14 from beyond the arc.
Phil Pressey scored 19 points and also had 19 rebounds for Missouri.