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...
9. Josh Brownhill 6
Off the pace in the first half, but able to plug the gap between the midfield and strikers following the introduction of Vydra and Rodriguez. Collected the ball in-between the lines to affect the game as Burnley heightened the pressure. Photo: Gareth Copley
10. Ashley Barnes 4
A nuisance physically, getting in front of his marker to hold up the ball and link play, but nothing really came off for the striker. Didn't see much/any of the ball inside the area and frustration crept into his performance. Booked before his withdrawal. Photo: Tim Keeton
11. Chris Wood 4
The striker looks starved of confidence at present. His supply chain wasn't great, but the New Zealander isn't doing enough on or off the ball to help himself. Shot wide of the near post in the first half and replaced by Rodriguez with 20 minutes left. Photo: Tim Keeton
12. Dwight McNeil 6
Made the difference when replacing Gudmundsson in the second half. Didn't give Coufal a moment's peace, driving at the full back repeatedly, and always finding a cross at the end of it. Hit the bar. A breath of fresh air for Burnley. Photo: Gareth Fuller