Aged 18 years and 21 days, the schoolboy batted for 599 minutes and faced 536 balls in an innings that included two sixes and 24 fours before he was bowled by Ryan Sidebottom.
He shared a 236-run stand with South African Hashim Amla (151) on Thursday.
Sibley's maiden first-class century arrived off 301 deliveries, making him the youngest Surrey player to reach that landmark. The previous record was set by Jack Crawford in 1905 at the age of 18 and 257 days.
The 200 came 183 balls later, with a boundary that brought the crowd to its feet and made Sibley the second youngest Englishman to score a first-class double century since WG Grace - who did it against Surrey at the Oval in 1866.
The previous record for the youngest 200 scorer was set by Northamptonshire's David Sales who made 210 against Worcestershire at the age of 18 and 237 days in 1996.
"I was so relieved to finally get my 100 but, when I went to 200, it was just unbelievable," Sibley, whose school gave him time off his studies for the match, told reporters on Thursday.
"I'm not sure what I would have been doing had I been at school today, hopefully it would have been a free period."
(Reporting by Alan Baldwin, editing by Ed Osmond)
