The Eagles (10-2) gained 477 of their 607 yards on the ground as the two teams totaled 1,167 yards of offense.
The Monarchs gained 560 yards with Taylor Heinicke throwing for 341 on 25 of 44 passes. He added 75 yards on seven carries. Old Dominion (10-3) gained the most yards allowed by Georgia Southern this season.
Tied at 28 at the half, Swope ran 76 yards on the first play of the third quarter. Old Dominion answered with a touchdown to tie it, but the Eagles took the lead for good on Nico Hickey's 13-yard TD run with 3:45 to go in the third.
Georgia Southern hosts Maine in next Saturday's quarterfinals.