Thomas scored 15 points with nine rebounds and Devin Mitchell made five 3-pointers for his 15 points for the Panthers (14-4, 5-0 Sun Belt).
Kane Williams gave Georgia State the lead with two free throws with 1:09 remaining and Thomas hit a 3-pointer 30 seconds later and added a free throw.
South Alabama had three shots at a tying 3-pointer in the final six seconds.
Georgia State led by six at halftime and went up by 11 with 17 minutes to go after a 9-0 run. Ajayi scored four straight points to give the Jaguars their only lead of the second half with 1:39 remaining.
Kenyon Martin's son commits to Vanderbilt
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2020 March Madness dates, NCAA tournament dates, sites, locations
The 2020 NCAA tournament field will be announced on Selection Sunday, March 15. The Final Four will take place in Atlanta from April 4-6.