The Tigers built an 11-point lead on a 31-yard Colby Delahoussaye field goal with 8 minutes left in the game, but Tyler Murphy then led the Gators to the LSU 20-yard line and was threatening to score with 4 minutes to go.
Mills, however, sacked Murphy on third-and-15 for a 12-yard loss, and on the next play Beckwith sacked him again and forced a fumble, which was recovered by D.J. Humphries to seal the victory.
LSU (6-1 SEC, 3-1) notched four sacks overall and limited the Gators to just six points and 239 yards of total offense.
Zach Mettenberger needed to make good on just 9-of-17 passes for 152 yards and Jeremy Hill picked up 121 rushing yards on 19 carries in the victory.
Florida (4-2, 3-1) saw its three-game winning streak snapped as Murphy was held to just 116 passing yards on 15-of-27 efficiency.