Goals from Alvaro Morata, Ross Barkley, Willian and Ruben Loftus-Cheek moved the Blues to within two points of leaders Liverpool, and earned Sarri a slice of history.
No Blues manager had ever avoided defeat for their first 10 Premier League games, with the former Napoli boss the first to do so.
Chelsea have won seven of their opening 10 games, beating Huddersfield, Arsenal, Newcastle, Bournemouth, Cardiff, Southampton and the Clarets, while they have also drawn three, against West Ham, Liverpool and Manchester United.
During that time, they have also remained unbeaten in their first three Europa League group games, and are through to the fourth round of the Carabao Cup, having beaten Jurgen Klopp's Reds in the previous round.
The only Premier League manager to have a better record from the beginning of their reign is Frank Clark, who went 11 games unbeaten between August and October 1994 at Nottingham Forest.
Chelsea have twice previously gone 10 games unbeaten at the start of a season, in 2005-06 and 2014-15. They won the Premier League title on both occasions, each under Jose Mourinho.
The game at Turf Moor was also notable for Barkley, who has now scored in three consecutive Premier League games for the first time in his career. He is the first Englishman to achieve that feat for Chelsea since Frank Lampard in 2013.
In addition, it was also the first time he had been involved in three goals for Chelsea as he scored once and provided assists for two more.
The Blues are next in action against Derby County in the EFL Cup, in a reunion with legendary midfielder Frank Lampard, who is now Rams manager.
They will then face Crystal Palace in the Premier League, before a Europa League trip to BATE Borisov in Belarus.