The Tigers and Cardinal met in the opening game of the Battle 4 Atlantis, an eight-team tournament featuring four clubs that are nationally ranked.
Bowers scored 19 points and grabbed 11 rebounds to keep Missouri unbeaten in its first four games of the season.
Missouri held a two-point lead with 3:15 to go, but pulled away down the stretch while handing Stanford (3-2) its second straight loss.
The Tigers shot 36 percent from the field, but grabbed 18 offensive rebounds and turned the ball over just 10 times.
Phil Pressey added 18 points with eight assists and five rebounds for the Tigers.
Stanford was led by Chasson Randle with 22 points and Dwight Powell with 18.