The Crimson Tide (9-0, 6-0) found themselves trailing by only three points with 1:34 remaining after LSU's Drew Alleman missed a 45-yard field goal.
That gave McCarron (14-of-27, 165 yards, one TD) a last-chance shot for a win or a tie. He hit Kevin Norwood on three consecutive passes -- moving the ball to the LSU 28 -- and one play later tossed a screen to Yeldon, who ran 28 yards into the end zone with 51 seconds left.
Damion Square sacked LSU's Zach Mettenberger to end the game and Alabama remained undefeated after a rematch of last year's national title game.
Mettenberger finished 24-of-35 for 298 passing yards and a touchdown for the Tigers (7-2, 3-2), who lost for the first time in 23 home contests.