Erling Haaland's first-half strike secured the victory for Pep Guardiola's side, leaving them top of the Premier League table ahead of Arsenal.
But the result means Scott Parker's Burnley side have just 20 points with four games remaining, an insurmountable deficit to safety.
The Clarets are 13 points from safety as Tottenham Hotspur currently occupy 18th place with 31 points, with West Ham narrowly out of the relegation zone on 33 points.
It is the fifth consecutive season that Burnley have either been relegated or promoted as they oscillate between the riches of the Premier League and the Championship.
Three wins from their opening nine games offered hope that they could join Sunderland and Leeds in bucking the trend of promoted sides being immediately relegated back to the second tier.
However, just one win in their 25 matches since has sealed their fate.
Parker has become a Championship specialist, also winning promotion in the past with Fulham and Bournemouth.
But the former Tottenham and Chelsea midfielder has consistently struggled at all three clubs to make an impact in the Premier League.
