The striker converted Charlie Taylor's cross in the 38th minute to net his 10th goal of the campaign and lift Burnley up to ninth in the Premier League.
Sean Dyche's side have now lost just once in 11 games, claiming 10 points from their last four fixtures, and kept a 14th clean sheet of the season.
Here's how we scored the players' performances...
9. Dwight McNeil 8
Had more to do defensively in Burnley's own third of the pitch in the first half and the latter stages of the second, but supported Taylor well. Stretched the play for the winning goal and provided an excellent outlet on the ball to drive his side forward.
10. Jay Rodriguez 8
Took his goal brilliantly, regardless of whether it came off his head or his shoulder, anticipating the cross and moving well to get ahead of Cresswell. Stretched West Ham's defence, chasing balls into the channels, and work hard to close the play.
11. Matej Vydra 7
A typically selfless display from the Burnley forward who ran himself into the ground at the London Stadium. Played on the shoulder, getting in behind on a number of occasions, and went close with a first half header from McNeil's set-piece.
12. Chris Wood 7
Vibrant after replacing Vydra on the hour, picking up some good positions and holding up the ball well. Denied a goal on his 100th Premier League appearance by VAR and tested Fabianski after getting in behind from a Taylor pass.