Resuming on 169, Sibley, who made the last of his 22 Test appearances for England four years ago, made a career-best 305, with the opener thumping 29 fours and two sixes in a marathon 475-ball innings that ended with Surrey on 745.
After Sam Curran's 108 on Sunday, Dan Lawrence made it to three figures with 178 off 149 balls, while Will Jacks also had licence to tee off and he thumped 119 off 94 deliveries at London's Oval.
Durham did not even get the consolation of bowling Surrey out as Rory Burns declared after Jacks was out but not before his side had eclipsed their highest-ever total of 811, which was set in 1899.
In reply, Durham were 1-59 at the close, with former Australian international Dan Worrall bowling six overs to help contain the visitors.
Elsewhere in the Championship, another Australian, Daniel Hughes' 151, helped Sussex to 3-278 after Warwickshire were all out for 415 at Hove.
The 36-year-old left-hander from Sydney enjoyed an opening partnership of 98 with Tom Haines before adding 142 with James Coles before falling to Aussie compatriot off-spinner Corey Rocchiccioli, off the penultimate delivery of the second day.
Durham need 671 just to avoid the follow-on against the reigning county champions and they ended day two on 1-59.
Worcestershire captain Jake Libby made 228 not out a day after Adam Hose's explosive 266 as Worcestershire racked up 7-679 against Hampshire.
Ben Slater underpinned Nottinghamshire's 3-214 with 116 not out in response to Somerset's 379 at Taunton, with Tom Banton's 84 the top score for the home side.
Essex slipped from 3-273 to 368 all out against Yorkshire, who went to stumps on 3-143 for three at York thanks to opener Adam Lyth's stoic unbeaten 65 from 172 balls.
In Division Two, England's record Test wicket-taker and Lancashire captain James Anderson collected 2-45 as Derbyshire were all out for 261 after the Red Rose had made 367 at Chesterfield.
Anderson struck in his second and third overs, snaring Mitchell Wagstaff lbw and then bowling Australian opener Caleb Jewell.
First-innings centurion Keaton Jennings made 51 not out in Lancashire reaching 1-114 and an overnight lead of 220.
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