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.
Georgia Southern (8-4) finished third in the nine-team Southern Conference and wasn't sure of its playoff fate. But the Eagles looked like contenders Saturday.
'We felt privileged and honored to be one of the 20 teams," Eagles coach Jeff Monken said. "We were going to make the most of the opportunity."
Shaw and his teammates gave glimpses of Georgia Southern teams that had playoff swagger. The Eagles drove 80 yards on 11 plays with their first possession, capping the march with Wilcox's 5-yard touchdown run.
Seven seconds later during a kickoff, Michael Butler tackled SCSU's Donovon Richards in the end zone for a safety to expand the score to 9-0.
After an exchange of punts, Georgia Southern was backed up to its 5. Shaw then absorbed the big hit.
But Southern maintained the momentum in Shaw's absence as the defense converted turnovers into points.
RICHARD BURKHART/MORRIS NEWS SERVICE
Georgia Southern's Brent Russell forces a fumble as S.C. State QB Malcolm Long tries to hand the ball off to Asheton Jordan.