Duane Notice added 11 points and Mindaugas Kacinas grabbed a game-best 10 rebounds for the 13th-seeded Gamecocks (14-19), who will play fourth-seeded Tennessee in a Friday quarterfinal.
Coty Clarke paced fifth-seeded Arkansas (21-11) with 13 points, while Rashad Madden and Bobby Portis each netted 11.
Thursday's tight second-round matchup featured 21 lead changes.
Brenton Williams made 1-of-2 free throws to put South Carolina up 70-69 with 53 seconds remaining. Madden missed a floater in the lane at the other end of the floor and the ball went out of bounds off an Arkansas player.
Thornwell was fouled with 20.8 seconds left. However, he missed both free throws after having been a perfect 6-for-6 at the line.
The Razorbacks then missed another opportunity to reclaim the lead when Portis couldn't connect with a hook shot down low.
Michael Carrera went to the line for South Carolina with 2.9 seconds to play and converted 1-of-2 attempts.
The Gamecocks prevailed when Madden's heave from just past the half-court line was off the mark.