Michail Antonio scored the only goal of the game in the ninth minute to wrap the three points up for the Hammers.
Pablo Fornals supplied the cross from the left hand side and the 30-year-old forward stole a yard from Robbie Brady to touch the ball home from close range.
The Clarets cranked up the pressure in the second half - with Dwight McNeil, Matej Vydra and Jay Rodriguez featuring from the bench - but they couldn't force an equaliser.
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The Clarets cranked up the pressure in the second half - with Dwight McNeil, Matej Vydra and Jay Rodriguez featuring from the bench - but they couldn't force an equaliser.