The 1-0 home defeat to Manchester City left Burnley with no chance of beating the drop.
The Clarets released a statement in the aftermath, saying: "Confirmation of relegation from the Premier League has been the consequence of a difficult season for everyone connected with Burnley Football Club.
"We began the campaign fully aware that the most demanding league in the world would present significant challenges. This outcome is not what anyone connected with the club wanted.
"This season has presented challenges that have tested every part of the club. There have been periods of progress and performances that demonstrate the standards we expect, but ultimately, we have fallen short of the quality and consistency required to compete at Premier League level.
"We would like to thank our supporters that have been with us through every high and low, home and away. That continued support is fundamental to everything we do.
"Burnley Football Club has 144 years of history. It has never been defined by a single season, but by its ability to respond and move forward."