The Crimson Tide (9-2) took on a lightly-regarded, non-conference opponent as a tuneup for their showdown with second-ranked Auburn next week, a game that will have national championship implications.
A win by Auburn in that game and another one in the SEC title contest will all but guarantee the Tigers will play for the national crown.
Alabama will try to deny its old rival that opportunity and prepared for that game with a 42-point first half.
McElroy hit Jones with an 8-yard touchdown pass on Alabama's first possession and found him again from 10 yards out in the second quarter to make it 28-0.
The Crimson Tide scored second-period touchdowns on C.J. Mosley's 42-yard interception return and Brandon Gibson's 36-yard blocked punt return.
Georgia State's only score came on a 97-yard kickoff return by Albert Wilson after Alabama had scored its sixth touchdown.
McElroy was 12-of-13 for 159 yards before giving way to backup A.J. McCarron. Georgia State fell to 6-5.