Trailing 24-22, Clemson (11-2) took possession on its 20 with 1:39 remaining. Tajh Boyd completed a pass for 26 yards to DeAndre Hopkins on a fourth-and-16 play during the decisive 10-play drive.
Catanzaro's kick set off a wild celebration on the field and in the stands. Some players collapsed on the field in apparent disbelief while most of Clemson's orange jerseys met in a midfield circle.
Hopkins, who had 13 catches for 191 yards and two touchdowns, also had receptions for 7 and 13 yards in the final drive. LSU safety Greg Reid was flagged for pass interference while defending Hopkins.
Jeremy Hill ran for 124 yards and two touchdowns for LSU (10-3), which carried a 24-13 lead into the final quarter.
Boyd completed 36 of 50 passes for 346 yards with two touchdowns and no interceptions. He set career highs for attempts and completions while winning the game MVP award.
LSU scored 10 points off Clemson's two lost fumbles, including one by Sammy Watkins on the second play of the game that set up Hill's first touchdown.
Hopkins scored on an 11-yard reception in the second quarter and a 12-yard grab in the fourth. LSU's Bennie Logan blocked Catanzaro's extra point attempt following Hopkins' first touchdown.