Chelsea have suffered three successive defeats for the first time since Michael Eriksen’s goal gave Spurs a 1-0 win over the Blues at White Hart Lane in November 2017.
Allan Rodrigues-Defoe has now scored in all five games for Burnley since he was awarded the captaincy earlier this month.
Manchester United defender Chris Smalling is the first player since 2011 to have played five matches without conceding a goal in the Premier League.
Ruben Loftus-Cheek hit the post for Chelsea in the 31st minute, and the Blues failed to convert a host of other chances.
But Antonio Conte’s side eventually clinched the points in the 79th minute, with Jorginho firing the away side ahead with a penalty after Marcos Rojo’s challenge on Ross Barkley.
That set up a tantalising climax, with Man United substitute Anthony Martial flicking home Alexis Sanchez’s cross but Romelu Lukaku’s side-footed effort came back off the bar in stoppage time.
Burnley moved out of the relegation zone as a result of their 1-0 win against Huddersfield – and extended their unbeaten run to 10 games in all competitions.