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.
Here are the ratings...
13. Jay Rodriguez 5
The striker will be pushing for a start after an encouraging cameo at the London Stadium. Stretched the home side's defence, offered more of a presence in the final third, and got across his marker repeatedly inside the box. Photo: Molly Darlington
14. Matej Vydra 5
Made a favourable impression once again after featuring from the bench. The Clarets looked far more vibrant in the final third following his introduction and the Hammers were made to dig in as a result. Needs to start. Photo: Molly Darlington