The Yellow Jackets (4-4) already had lost to UNC-Greensboro and Winthrop, and trailed the lowly Panthers 40-32 after going nearly 5 minutes without a point to start the second half.
But Georgia State (2-5), which was playing host to Georgia Tech for the first time since 1973 even though the Atlanta schools are only a couple of miles apart, couldn't hold on for the first signature win of coach Rod Barnes' tenure.
Clinch hit a jumper in the lane with 6:02 remaining to cap a 13-5 run that gave the Yellow Jackets their first lead of the second half, 58-56.
Leonard Mendez, who led the Panthers with 20 points, made two free throws to tie the game again, but Josh Peacock scored on the inside after setting a big pick near midcourt. Georgia Tech led the rest of the way.
Georgia State's last chance came after Peacock was whistled for a technical with 2:01 remaining. Mendez made two free throws to pull the Panthers within 64-62, and they got the ball back with 6 seconds left on the shot clock.
But Rashad Chase stumbled trying to drive and Georgia State never got off a shot.
Clinch wrapped the contest up at the foul line, hitting a pair of free throws with 3.7 seconds remaining to finish off the Panthers.
Clinch was 6 for 14 from the field and hit all six of his free throw attempts.
Peacock added 17 points for Georgia Tech.
The score was tied 32-32 at halftime, thanks to Georgia State's Trae Goldston, who hit a 3-point shot with 23 seconds left before the break.