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.
Members of Congress to keep receiving porn magazine
Putin thinks Obama would save him if he were drowning