After Andre Stringer had given LSU the lead on a layup with 15 seconds left in the extra time, Kentucky's James Young drove the lane as the clock wound down, but lost the ball.
Randle corralled it and converted the game-winner with a hook shot.
Stringer committed a turnover on the Tigers' final possession and the Wildcats (21-6, 11-3 SEC) hung on for their sixth win in seven games.
Aaron Harrison scored 21 points and Young ended with 20. Randle had eight points and 15 rebounds for Kentucky.
Johnny O'Bryant III had 20 points and 12 rebounds for LSU (16-10, 7-7). Anthony Hickey scored 20 points.