Erling Haaland opened the scoring after five minutes to settle the contest. Burnley were unable to find an equaliser despite restricting City to a single goal from 27 attempts, and their fate was confirmed at the final whistle.
It is the third Premier League relegation in five seasons for the Lancashire club, who have consistently oscillated between the divisions during a turbulent period in their recent history.
Scott Parker's side have picked up just 20 points all season, three above already-relegated Wolverhampton Wanderers at the bottom of the table.
A run of one win in 25 matches since a promising start to the campaign ultimately made the outcome inevitable.
The result also sends Manchester City to the top of the Premier League table ahead of Arsenal on goal difference, with five games left for both sides in the title race.
Burnley released a statement following the final whistle: "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."
The club added: "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."