Chelsea had captain Gary Cahill and Cesc Fabregas sent off and Burnley took full advantage through Sam Vokes's brace and a superb Stephen Ward effort, with goals from Alvaro Morata and David Luiz in vain.
"We tried until the end to reach a good result against Burnley. It was a pity because our fight in the second half was very good," said Chelsea manager Antonio Conte.
"But our first half, we kept losing our heads. After the red card we lost our heads, conceded two goals and then it's very difficult to change the final result."Conte was without injured Eden Hazard and out-of-favour Diego Costa, while £58-million signing Morata was on the bench.
Youth team graduate Jeremie Boga made a debut, but was hauled off in the reshuffle after Cahill's 14th-minute dismissal for a reckless, studs-up foul on Steven Defour.
Vokes scored in the 24th minute, volleying in Matthew Lowton's cross. Ward doubled the advantage in the 39th minute with a blistering left-foot strike from Jack Cork's lofted pass and Vokes added a third before half time, heading in Defour's cross.
Half-time substitute Morata and Luiz reduced the arrears in the second half, either side of Fabregas's 81st-minute dismissal for a second yellow card, but Burnley hung on.
"Did we expect this situation? No, if we're honest," said Burnley manager Sean Dyche.
"It was a fantastic three points and the referee, I just want to say, was fantastic."
Manchester City manager Pep Guardiola had £150-million of recruits on display at Brighton but it took the favourites until the 70th minute to take the lead.
David Silva fed Sergio Aguero to run through and beat goalkeeper Mathew Ryan and five minutes later Lewis Dunk headed Fernandinho's cross into his own net...