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...
5. Ben Mee 6
Will be disappointed in the role he played in West Ham's opener. However, he did enough to prevent Ogbonna from making it 2-0 in the first half, he cleared Antonio's header off the line after the break and made a tremendous challenge to deny Yarmolenko. Photo: Catherine Ivill
6. Johann Berg Gudmundsson 4
Forced Fabianski into an early save, but couldn't find his way into the game at any point. Put the hard yards in, but didn't do enough to press the ball when Fornals supplied the cross for Antonio to score. Photo: Andrew Boyers
7. Robbie Brady 5
Culpable for West Ham's winner when failing to get goal side of Antonio. No excuses for the part he played in that. His distribution improved in the second half, but his delivery from dead ball situations requires vast improvement. Photo: Gareth Copley
8. Ashley Westwood 6
Distribution was mixed in the first half and Burnley's inability to keep the ball, in the first half in-particular, gave Rice the edge. Picked up after the break, dictating play, and winning the ball high up the pitch to pin West Ham back. Photo: Gareth Copley