High-flying Millwall dent Burnley's Premier League promotion push
Scott Parker's side would have gone second in the Championship with a victory.
Instead, they left south London empty-handed after Millwall captain Cooper headed the second half winner.
Millwall's third successive victory lifted them into seventh place, while Burnley are fourth following the end of their nine-match unbeaten run.
A lack of goals has cost Burnley this season and when Luca Koleosho crashed an angled drive against the near post after only five minutes, it was the closest Parker's team came to scoring all match.
Burnley striker Andreas Hountondji threatened with a shot that fizzed narrowly past the post from 20 yards.
Millwall grew into the game but Cooper could not direct his header from Femi Azeez's cross inside the far post and Macaulay Langstaff volleyed over.
A cross from Casper De Norre was headed goalwards by Longstaff, forcing a fine save from Clarets keeper James Trafford.
From the corner, Azeez volleyed over when he should have hit the target.
Neil Harris's men carried that momentum into the second half and got their reward in the 52nd minute.
Azeez teased Josh Brownhill down the right flank before lofting a cross into the Burnley area, where Cooper rose highest to power a towering header into the top corner.
