Murray completed 22-of-35 passes while Todd Gurley compiled 65 rushing yards and a touchdown on 10 carries for the Bulldogs (2-0, 1-0 SEC).
Ken Malcome added a TD run and Marlon Brown racked up 106 yards on eight receptions for the winners.
The Tigers (1-1, 0-1) fell in their first-ever contest as a member of the Southeastern Conference despite 269 passing yards from James Franklin.
The Missouri signal-caller completed 25-of-41 passes for 269 yards, two touchdowns and an interception.
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