Gillislee rushed for 146 yards and two second-half touchdowns in Florida's 14-6 win over LSU.
Elam forced a fumble late in the fourth quarter, and Minter had a school record 17 solo tackles.
South Carolina return man Ace Sanders and Vanderbilt punter Richard Kent shared special teams honors. Sanders scored on a 70-yard punt return against Georgia. Kent had a net average of 44.9 yards on six punts against Missouri.
Gamecocks center T.J. Johnson and Arkansas defensive end Trey Flowers are linemen of the week.
Mississippi State linebacker Bernardrick McKinney and Texas A&M quarterback Johnny Manziel are co-freshmen of the week.