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.
After Hopkins' second TD catch, LSU got the ball with 2:43 remaining and threw three straight passes. Only one was complete in the three-and-out series that took only about one minute off the clock, leaving Clemson with enough time for its winning drive against LSU's exhausted defense.
Hill did not have a carry in the fourth quarter. Zach Mettenberger was sacked six times.
"We had to throw the football," LSU coach Les Miles said.
"It's a tough thing to figure our pass protection was as poor as it was. That's the piece that needs fixing and frankly we've got some young players here we'll address and improve and make some strides going forward."
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.
Clemson had a chance to tie it after Hopkins' second TD, but Boyd's pass for the 2-point conversion was incomplete.
Michael Ford had a 43-yard kickoff return for LSU to open the second half. On first down, Hill broke through the line for a 57-yard touchdown run. His 12th rushing touchdown broke Dalton Hilliard's LSU record for a freshman set in 1982.
Clemson lost Watkins to a right ankle injury on his early fumble. Watkins, a wide receiver, took the handoff as a running back and fumbled when he was hit by defensive end Barkevious Mingo. Safety Craig Loston recovered at the Clemson 23.
LSU needed only two runs by Hill to move in front. His 17-yard scamper made it 7-0.
Watkins could not put pressure on his right foot as he was helped off the field and then driven to the locker room. Clemson announced X-rays were negative but Watkins would not return. No further details were released.
Clemson already was without backup receiver Martavis Bryant, who was suspended for the game for failing to meet academic requirements.
Clemson's second costly fumble came midway through the third quarter. Running back Andre Ellington ran for 8 yards but lost the ball when hit by defensive end Sam Montgomery. Reid recovered the fumble at the Clemson 29, setting up Drew Alleman's 20-yard field goal.
The injury to Watkins left the spotlight on Hopkins, Clemson's leading receiver. He had catches of 17 and 12 yards as Clemson pulled even with an 11-play drive capped by Boyd's 11-yard touchdown run.
After LSU regained the lead on Mettenberger's 6-yard touchdown pass to Jarvis Landry, Hopkins had a 31-yard catch to set up his 11-yard score late in the second quarter. Hopkins' sliding grab gave him TD receptions in 10 straight games to set an Atlantic Coast Conference record. Virginia's Herman Moore had touchdown catches in nine straight games in 1990.
Mettenberger completed 14 of 23 passes for 120 yards with an interception.