Catanzaro's 37-yard kick with 2 seconds remaining came after Tajh Boyd led the Tigers (11-2) on a 10-play, 60-yard drive over the final 1 1/2 minutes, highlighted by a 4th-down, 26-yard completion to DeAndre Hopkins.
The game-winning kick capped a run of 12 straight fourth-quarter points that rallied Clemson from a 24-13 deficit.
Boyd finished 35-of-49 through the air for 339 yards with a pair of touchdowns to Hopkins, who amassed 190 yards on 13 grabs.
The Tigers signal-caller also carried the 29 times for 22 yards and a score in the victory.
Zach Mettenberger completed 14-of-23 passes for 109 yards, a touchdown and a pick-off for LSU (10-3).
Jeremy Hill racked up 125 yards and scored twice on 12 carries in the Tigers' loss.