The Volunteers had the ball first in the extra period, but Alton Howard fumbled out of the end zone, allowing Aaron Murray to march the Bulldogs (4-1, 3-0 SEC) 42 yards on four plays, capped by Morgan's game-winning kick.
Georgia had forced overtime when the senior signal-caller drove 75 yards on 10 plays in the final 2 minutes of regulation, capped by a 2-yard TD pass to Rantavious Wooten with 5 seconds remaining.
Murray finished 19-of-35 for 196 passing yards and three touchdowns overall, becoming the SEC's all-time leader in passing yards with 11,625.
Wooten made six grabs for 38 yards with two scores and J.J. Green ran for 129 yards on 17 carries.
Justin Worley made good on 17-of-31 passes for 215 with a TD for Tennessee (3-3, 0-2).
Rajion Green racked up 148 rushing yards with two scores on 28 carries for the Vols.
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