McCarron racked up 183 yards on 22-of-30 passing and wasn't intercepted for the Crimson Tide (5-0, 2-0 SEC).
Cooper made eight grabs in all for 85 yards. Eddie Lacy ran the ball 19 times for 81 yards for the winners.
The Rebels (3-2, 0-1) were paced by Bo Wallace, who went 15-of-26 through the air for 123 yards with two interceptions.
Celebrity Families of 2014 [PHOTOS]