Kyle Fuller added 14 points for Vanderbilt (2-3). The Commodores, looking to replace their top six scorers from last season's 25-win team, entered averaging 55.8 points a game.
John Bohannon had 11 points for the Miners (1-4), who have lost four in a row. The streak matches their longest skid under coach Tim Floyd.
Both teams lost their previous two games at the classic.
Vanderbilt was coming off a 50-33 loss Friday to Marist. The Commodores were also defeated 75-62 by Davidson. UTEP's losses came against Clemson (69-48) and Oklahoma (68-61).
Johnson's 3-pointer made it 57-36 with 7:30 to play. The guard, who had a career-high 28 points in Thursday's loss to Davidson, was held to four points one day later by Marist.
Vanderbilt opened the second half with a 9-2 run to grab a 42-19 lead. Shelby Moats had a 3 during the stretch.
UTEP narrowed the gap to 49-34 midway through the second on C.J. Cooper's basket.
Odom had eight points in helping Vanderbilt go ahead 33-17 at halftime.
Kevin Bright had two 3-pointers, and Odom and Fuller also hit long-range jumpers during a 19-0 stretch as Vanderbilt took a 28-12 advantage 14 minutes into the game.
UTEP went scoreless for more than eight minutes before Cedrick Lang's basket cut the deficit to 28-14 four minutes before intermission.
Lang finished with 10 points.
Bohannon's 3-point play gave UTEP an early 7-1 lead. Odom made a jumper to conclude an 8-0 run for Vanderbilt that put the Commodores up 9-7.