Match ends, West Ham United 0, Burnley 1.
Jay Rodriguez's winner edged Burnley closer to a Europa League spot as relegation-threatened West Ham were left to rue a host of missed chances.
The forward met Charlie Taylor's whipped delivery with a flicked header that looped over Lukasz Fabianski and dropped in off the crossbar.
It came after a string of fine saves from Clarets goalkeeper Nick Pope.
Michail Antonio also struck a post and Pope then turned substitute Sebastien Haller's effort wide with his studs.
It was a glorious chance with his first touch of the game for West Ham's record signing, who came off the bench after returning from a hip injury.
Chris Wood thought he had added a second for the visitors with 14 minutes to go when he tapped in Phil Bardsley's low cross, but was ruled offside.
The win takes Sean Dyche's side to within one point of eighth-placed Arsenal, while the Hammers remain four points above the relegation zone.
In-form Burnley eyeing Europe
With injuries mounting, the departure of out-of-contract players and their Premier League status safe, Burnley could have been forgiven for easing their way to the end of the season.
Instead Dyche's side are one of the division's most in-form teams and eyeing a Europa League spot that seemed out of their reach before Christmas.
Burnley have lost just once in 12 top-flight games and a clean sheet at London Stadium means they now have a league-high 14 to their name this term.
Rodriguez took his first-half winner superbly, one of few chances that came his way, but Dyche had goalkeeper Pope to thank for the three points as the England international made a number of smart saves.
He snuffed out West Ham's early pressure with a good block from Tomas Soucek's close-range volley and took command of his box as the hosts threw bodies into the Burnley penalty area late on.
Dyche was able to name a full complement on his bench, having twice been unable to fill the nine spots since the restart, and welcomed the return of forward Wood from injury.
The New Zealand international, after having an effort ruled out for offside, provided a valuable outlet for the visitors and forced Fabianski into a save as Burnley looked for a second on the break.
However, the visitors were perhaps fortunate to not be reduced to 10 men after James Tarkowski's dangerous tackle on Jarrod Bowen, instead the Clarets skipper escaped with a yellow card.
"It's a red card," said Hammers boss David Moyes afterwards. "It's not going to change the result - it might have done during the game.
"It is really disappointing that the referee didn't see it as a red, the linesman is very close to it and we have the chance to go to VAR to make a decision on it."
Crunch time for Hammers
This was the second of three games in seven days for the Hammers, a run of fixtures Moyes bemoaned earlier this week and will not be any happier about after losing a contest they dominated for large spells.
The hosts had 22 shots but only managed five on target, one of which saw £45m striker Haller denied by Pope seconds after coming off the bench.
Antonio and Bowen linked up well in the first half to cause Burnley problems while on-loan midfielder Soucek would have opened the scoring if his volley was not straight at the goalkeeper.
And despite throwing bodies forward in search of a leveller, the hosts created little of note in the second half.
West Ham's hectic spell will be completed by a visit to Norwich on Saturday, when a win at Carrow Road would not only boost their own survival hopes but also relegate their hosts.
Ultimately, the Hammers' destiny is in their own hands with home games against relegation rivals Watford and Aston Villa either side of a trip to Manchester United also still to come.
But they will watch on a little less comfortably when Bournemouth and Villa look to close the gap in their games in hand on Thursday.
Man of the match - Nick Pope (Burnley)
'We have a culture we believe in'
Burnley boss Sean Dyche: "Last few games have been fantastic. It's a challenge when they lose team-mates. There's a resilience to them, you add in organisation and physical prowess, and they've got more talent than people mention.
"Two centre halves were outstanding, the keeper was outstanding. They asked a lot of questions, had a couple of golden moments, and our keeper was there when needed.
"We have a culture and environment we believe in. It's that commitment to the cause. The next game comes very quickly and it's Liverpool, they're not bad…"
West Ham boss David Moyes: "Frustration is probably the word I would use more than anything, the missed opportunities. I am disappointed that we didn't take something from the game because I feel we warranted something.
"I thought today was a must-win, I thought Newcastle was too, Chelsea we had to do the best we can and we did. We have been in that situation for a while. In fact, probably since I took over."
Clarets enjoy life on the road - the stats
- Burnley have picked up 13 points from the last 18 available to them in away league games, as many as they earned across their previous 16 such matches on the road beforehand.
- This was West Ham's 19th Premier League loss of the season; only in 2006-07 (21) and 2013-14 (20) have the Hammers suffered more defeats in a single campaign.
- Burnley have won back-to-back away Premier League games without conceding for the first time since May 2015.
- West Ham have lost three consecutive league meetings with Burnley for the first time since May 1947.
- Burnley have won successive away Premier League games in London for only the second time, having previously done so back in August 2017.
- The Clarets have kept a clean sheet in both of those wins, having only kept two shutouts in their previous 24 Premier League away matches at London sides before this.
- Burnley's Ja Rodriguez scored his seventh league goal of the season; only in 2013-14 for Southampton (15) has he ever scored more in a single Premier League campaign.
West Ham visit Norwich at Carrow Road on Saturday (12:30 BST), while Burnley head to champions Liverpool later the same day (15:00).
- 24FredericksBooked at 64mins
- 7YarmolenkoSubstituted forHallerat 62'minutes
- 18FornalsSubstituted forAjetiat 86'minutes
- 5TarkowskiBooked at 58mins
- 27VydraSubstituted forWoodat 58'minutes
- Michael Oliver
- Shots on Target
Second Half ends, West Ham United 0, Burnley 1.
Foul by Declan Rice (West Ham United).
Dwight McNeil (Burnley) wins a free kick in the attacking half.
Attempt missed. Albian Ajeti (West Ham United) header from outside the box misses to the right. Assisted by Angelo Ogbonna with a cross.
Attempt blocked. Michail Antonio (West Ham United) header from the centre of the box is blocked. Assisted by Jarrod Bowen with a cross.
Corner, West Ham United. Conceded by Josh Brownhill.
Corner, West Ham United. Conceded by Charlie Taylor.
Attempt blocked. Jarrod Bowen (West Ham United) left footed shot from outside the box is blocked. Assisted by Ryan Fredericks.
Offside, Burnley. Chris Wood tries a through ball, but Jay Rodriguez is caught offside.
Substitution, West Ham United. Albian Ajeti replaces Pablo Fornals.
Attempt missed. Ashley Westwood (Burnley) right footed shot from the left side of the box is close, but misses to the right. Assisted by Jay Rodriguez.
Attempt saved. Chris Wood (Burnley) right footed shot from the left side of the box is saved in the bottom right corner. Assisted by Charlie Taylor.
VAR Decision: No Goal West Ham United 0-1 Burnley.
Offside, Burnley. Phil Bardsley tries a through ball, but Chris Wood is caught offside.
Corner, West Ham United. Conceded by Charlie Taylor.
Attempt blocked. Jarrod Bowen (West Ham United) left footed shot from the centre of the box is blocked. Assisted by Tomas Soucek.
Lukasz Fabianski (West Ham United) wins a free kick in the defensive half.
Foul by Kevin Long (Burnley).
Corner, Burnley. Conceded by Issa Diop.