Gary Cahill was shown a straight red for a lunging tackle on Steven Defour midway through the first half.
Burnley then capitalized with a ruthless display that saw them score 3 times before the break.
The Blues were better after the break and pulled a goal back when summer signing Alvaro Morata headed in shortly after coming off the bench.
Cesc Fabregas was then sent off for a late challenge on Jack Cork but David Luiz added 1 more goal for Chelsea.
Elsewhere, Miguel Britos scored a dramatic 93rd-minute equaliser for Watford as Liverpool were held to a thrilling 3-all draw at Vicarage Road.
Wayne Rooney marked his top-flight return to Everton after 13 years by scoring the winner against Stoke and his 199th goal in the Premier League.
Huddersfield Town made a dream start to their debut season in the Premier League as they beat Crystal Palace 3-nil away.
Southampton were held to a goalless draw by Swansea at St Mary’s Stadium, as West Brom beat Bournemouth 1-nil .