Murray led the Bulldogs (3-1, 2-0 SEC) on a six-play march as the clock wound down, hitting Justin Scott-Wesley on a 25-yard scoring toss with 1:47 remaining -- his fourth TD pass of the game -- erasing a four-point deficit.
The Georgia senior completed 20-of-34 throws for 298 yards with an interception. He also ran for a touchdown.
Chris Conley amassed 112 receiving yards with a score on five catches while Michael Bennett hauled in two of Murray's TD throws.
Keith Marshall ran for 96 yards on 20 carries as Georgia notched a third straight win.
Ex-Bulldogs quarterback Zach Mettenberger had earlier put LSU (4-1, 1-1) in front with a late TD drive of his own, capped by a Jeremy Hill run with 4:14 left.
He went 23-of-37 passing for a personal-best 372 yards with three touchdowns.
Jarvis Landry led the Tigers' receiving corps with 156 yards on 10 grabs with a touchdown.