Bradley hooked up with Voisin for a 41-yard touchdown on the first possession for South Alabama (8-2, 5-1 Sun Belt Conference) and Diego Guajardo kicked a 35-yard field goal to put the Jaguars up 10-0 after one quarter.
Bradley hit DJ Thomas-Jones for a 2-yard touchdown in the second quarter and a 17-0 lead at halftime.
Layne Hatcher got Texas State (3-7, 1-5) on the scoreboard midway through the third quarter on a 4-yard scoring toss to Rontavius Groves, but Bradley answered with a 41-yard scoring strike to Caullin Lacy and the Jaguars led 24-7 with one quarter to go.
Bradley and Voisin teamed up for a 28-yard touchdown in the fourth quarter and Ke'Shun Brown capped the scoring with a 15-yard pick-6.
Bradley completed 20 of 34 passes for 274 yards for South Alabama. Voisin hauled in six passes for 118 yards.
Hatcher completed 24 of 42 passes for 143 yards for the Bobcats.