Frank Holmes scored 13 points to round out the double digit scoring for UAB (12-12, 4-6 Conference USA). He also grabbed 10 rebounds.
The Blazers broke a three-game losing streak at home with the win. UAB had previously won 19 straight home games before the skid.
Holmes powered inside for 10 points to lead UAB to a 40-26 halftime lead.
Kinnard scored four straight baskets early in the second half as the Blazers stretched their lead to 26 points at 54-28.
The game tightened down the stretch as Jackson and Kilgore heated up for UTEP and sparked a late rally. A driving basket by Jackson pulled the Miners to within 68-60 with 1:21 left.
UAB contributed to the UTEP comeback with poor shooting from the free throw line. They at one point made just 3 of 10 and were 12-of-30 for the game.