The Tigers (10-1, 6-1 SEC) had coughed up a 20-point lead in the fourth quarter and were trailing 38-37 lead entering the game's final minute.
On the winning play, Marshall was facing a fourth-and-18 from his own 27-yard line when he unloaded the ball down the middle toward Louis, who was double-teamed.
Bulldogs safety Tray Matthews touched it first, knocking it into the air, but Louis then hauled it in and ran untouched into the end zone.
Marshall completed 15-of-26 passes for 229 yards and added two rushing touchdowns for Auburn, which notched a seventh straight victory and set up a winner-take-all showdown with No. 1 Alabama for the SEC West title in the annual Iron Bowl game.
Louis made four grabs overall for 131 yards.
Tre Mason added a TD on 27 carries for 115 yards.
Georgia (6-4, 4-3) was led by senior quarterback Aaron Murray, who racked up 415 yards with two scores and one interception on 31-of-46 efficiency.
He also ran for a go-ahead touchdown in the fourth quarter.