Stein's third field goal in the third quarter put the Golden Eagles (6-3, 4-1 Conference USA) on top 23-2 before Jack Abraham had a pair of touchdown passes, 20 yards to Quez Watkins and 9 to Tim Jones.
Abraham was 17 of 22 for 164 yards and with Kevin Perkins picking up 94 yards on just six carries, Southern Mississippi piled up 325 total yards. UAB was leading the league in allowing just 255 yards, 92 on the ground.
A first-quarter interception by Shannon Showers led to a 6-yard touchdown run Steve Anderson in the first quarter. Thomas followed up Stein's first field goal with his pick-6 early in the second. Stein kicked a 44-yard field goal on the last play of the first half for a 20-2 lead.
UAB (6-3, 3-2) only had 173 yards on offense.
It was the 600th win in 103 years of football for Southern Mississippi.
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