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Dwayne Davis added 12 points to go with seven rebounds, Neil Watson added 11 points and Jonathan Mills chipped in 10.
UAB (8-10, 0-3) led 18-11 midway through the first half before a 10-2 spurt put Southern Miss on top 21-20 with just under 6 minutes left. From there, Southern Miss outscored UAB 17-7 to take a 38-27 into the break.
A Rashad McGill 3-pointer made the score 56-37 and Jenkins' trey with 5:30 showing gave the Golden Eagles a 20-point advantage, 66-46.
Southern Miss (15-4, 4-0), which entered the game ranked second nationally with 11.8 steals per game, forced 20 turnovers on the strength of 15 steals.
Terence Jones led UAB with 15 points.