Shaw was injured after a hit midway through the first quarter, but the Eagles found ample volunteers to step up during a 41-16 victory at Paulson Stadium.
"This team is very special," Georgia Southern backup quarterback Jerick McKinnon said. "There's a brotherhood here. We all believe in each other, and we'll fight for each other, and that's going to take you a long way."
With their first playoff win since 2002, the Eagles advanced to the round of 16. Southern will play second-seeded William & Mary in Williamsburg, Va., at 1:30 p.m. next Saturday.
The extent of Shaw's injury wasn't known after the game. Shaw was hit in the chest by South Carolina State's blitzing strong safety Christian Thompson while attempting an option pitch and was later taken to the locker room.
That left more than three quarters of the game in McKinnon's hands.
The true freshman started just one game this season, a loss to Samford.
But the Eagles took advantage of a mistake-prone South Carolina State team.
The Bulldogs (9-3) fumbled two kickoffs for safeties. Georgia Southern's defensive end Josh Gebhardt, who replaced injured Dion DuBose two weeks ago, returned a fumble for a touchdown and had an interception that set up a 19-yard touchdown drive.
The McKinnon-led Eagles piled up 323 rushing yards.
South Carolina State entered the game allowing an FCS-leading 70.4 yards on the ground per game.
The Eagles also counted on their reliable scorers. J.J. Wilcox ran for two touchdowns, and Adrian Mora booted field goals from 32, 23 and 45 yards.
And reserve quarterback Russell DeMasi added a fourth-quarter, 1-yard touchdown run to seal the win for the Eagles, who improved to 8-4.