The score was tied 17-17 early in the third quarter before the Crimson Tide (9-0, 6-0 SEC) reeled off the game's final 21 points to win going away.
Yeldon racked up 133 yards on 25 carries, including a 4-yard scoring run to put Alabama ahead for good. He tacked on a 1-yard TD in the fourth quarter.
AJ McCarron completed 14-of-20 passes for 179 yards and three touchdowns as the Tide notched a third straight win over LSU.
The Tigers, (7-3, 3-3) were paced by the passing of Zach Mettenberger, who went 16-of-23 for 241 yards and a touchdown.
Jeremy Hill carried the ball 13 times for 42 yards with a score.
LSU fell to 2-11-1 all-time against No. 1 ranked teams.