West Ham missed the chance to climb further away from relegation trouble after Jay Rodriguez fired Burnley to a 1-0 Premier League victory.
Rodriguez's first-half goal, a bullet header in off the underside of the crossbar, boosted Burnley's hopes of a place in Europe next season.
The Hammers could not find a way through a resolute Clarets defence after the interval and remain four points clear of the bottom three.
Visitors Burnley should have taken the lead when Matej Vydra collected Charlie Taylor's cross on the six-yard line and laid the ball off to Erik Pieters, but the Dutchman blazed over.
Vydra then met Ashley Westwood's free-kick with a glancing header which flew straight into the arms of Lukasz Fabianski.
West Ham began to look threatening, though, and Nick Pope had to block a volley from Tomas Soucek and keep out Michail Antonio's angled drive with his feet.
So it was against the run of play when Burnley took the lead in the 39th minute, Rodriguez darting in front of Aaron Cresswell and meeting Taylor's cross with a fine header for his 10th goal of the season.
The ball bounced down off the underside of the crossbar twice before finally spinning away from the flailing Fabianski and rolling over the line.
Antonio had the chance to equalise immediately but blasted his shot into the sidenetting before Pope saved from Andriy Yarmolenko low down at his near post.
James Tarkowski could count himself lucky to stay on the pitch early in the second half after planting his studs into the thigh of Jarrod Bowen.
It was a similarly ugly challenge to the one which saw Arsenal's Eddie Nketiah sent off 24 hours earlier, but referee Michael Oliver showed only a yellow card and the VAR chose not to intervene.
Hammers boss David Moyes sent on West Ham's record signing Sebastien Haller, fit again after a hip injury which had kept him out since the restart.
Within seconds the French striker had a glorious opportunity to equalise, but showed his ring rust with a weak finish too close to Pope.
Burnley had the ball in the net again when Chris Wood slid in Phil Bardsley's cross but he was flagged offside.
West Ham poured forward in search of an equaliser but Burnley held firm to stay on the fringes of the European places and keep the Hammers sweating for a little longer at least.