Diego Guajardo's 34-yard field goal was the only score in the first quarter as South Alabama took a 3-0 lead.
The Panthers (7-4, 4-3 Sun Belt Conference) took the lead for good in the second quarter on Destin Coates' 6-yard TD run and Ellington's 5-yard TD toss to Devin Gentry.
Barnett's 2-yard TD run was the only score in the third quarter and stretched Georgia State's lead to 21-3.
Ellington's 4-yard scoring toss to Roger Carter pushed the Panthers' lead to 28-3 just 7 seconds into the final period.
Ellington finished 21-of-28 passing for 208 yards with one interception for Georgia State. Carter finished with seven catches for 87 yards.
Trotter completed 12 of 25 passes for 218 yards with a pick.