After the Tigers had forged a 28-28 tie with 32 seconds left on Nick Marshall's 39-yard touchdown pass to Sammie Coates, the Crimson Tide took the ball to the Auburn 38-yard line for a 57-yard field goal try by Adam Griffith.
The kick was wide right, and Davis grabbed the ball on the fly about 6 yards deep in his own end zone.
He used a convoy of blockers up the left sideline to bolt to the game-winning touchdown in one of the most memorable plays in many years as the Tigers (11-1, 7-1 SEC) kept their national title hopes alive in the annual Iron Bowl game.
The victory -- which came a week after Auburn pulled off an unlikely win against Georgia -- earned the Tigers a spot in next week's SEC championship game.
The Crimson Tide (11-1, 7-1) saw their winning streak end at 15 games in heartbreaking fashion.