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.
Bets to keep an eye on for the NCCA tourney on Thursday
Doug Kezirian, Preston Johnson, Stanford Steve and Chris Fallica reveal their best bets for Thursday's slate of games in the NCAA tournament.
Who has the best hair in the NCAA tournament?
Here's our starting five. Who you got?
Wooden Watch: Could a contender's run this weekend upset Zion's chances?
Zion Williamson still feels like a sure choice for the Wooden Award, but a strong run this weekend from other contenders could sway voters.