Boyd said having 1:39 to stage the final drive "is like having 10, 15 minutes for us."
"When I saw that clock and that we had three timeouts I said 'Let's get it.'"
LSU quarterback Zach Mettenberger was sacked six times and completed 14 of 23 passes for 120 yards. He also threw for a touchdown and an interception.
"It's a tough thing to figure our pass protection was as poor as it was," Miles said. "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."
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. X-rays were negative.
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. Andre Ellington ran for 8 yards but lost the ball on a 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.