With the game tied late in the fourth quarter, Vanderbilt's Cassen Jackson-Garrison fumbled and Georgia recovered on its own eight-yard line with just under three minutes left.
The Bulldogs marched to the Vanderbilt 19-yard line before Coutu booted the game-winning field goal.
Knowshon Moreno had a career-high 157 yards rushing for the Bulldogs (5-2, 3-2 SEC), who scored 13 unanswered points in the second half and won for the fourth time in their last five games.
Matthew Stafford completed 16-of-31 passes for 201 yards and a touchdown while Tony Wilson had four catches for 57 yards in the win.
Mackenzi Adams threw for 125 yards and a touchdown on just 7-of-10 passing for the Commodores (3-3, 1-3), who have lost two straight. Adams also ran for 46 yards.