Managerless Burnley began life after Sean Dyche with a 1-1 draw at West Ham after Clarets winger Maxwel Cornet missed a first-half penalty at the London Stadium.
The relegation-threatened side, who sacked Dyche on Friday after a decade in charge, took the lead through Wout Weghorst (33), just three minutes after midfielder Ashley Westwood had been stretchered off following a serious injury that caused a lengthy delay.
Cornet then missed his penalty in stoppage time in the first half after he was brought down by West Ham goalkeeper Lukasz Fabianski when through one-on-one.
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The Ivorian was made to pay for his miss when Tomas Soucek equalised in the 74th minute, shouldering in Manuel Lanzini's free-kick.
Burnley, under the temporary charge of U23 coach Mike Jackson, held on thanks to two great saves from goalkeeper Nick Pope.
The Englishman denied Issa Diop and Michail Antonio late on to ensure Burnley moved to within three points of safety behind 17th-placed Everton.
West Ham, meanwhile, missed the chance to go fifth and remain seventh, five points behind fourth-placed Tottenham, who have a game in hand on the Hammers.
Pope spares Cornet's blushes
Having sacked Dyche just two days before their visit to West Ham, temporary boss Jackson, alongside injured club captain Ben Mee, who was part of the coaching staff at the London Stadium, was given the task of trying to guide Burnley to just their second away win of the season.
West Ham, though, buoyant from reaching their first European semi-final in 46 years on Thursday, started well as Pope was forced into a fine save at full stretch to deny Jarrod Bowen in the 13th minute, with Burnley skipper James Tarkowski clearing Aaron Cresswell's volley off the line four minutes later.
Burnley's Westwood then suffered a serious injury in the 22nd minute following a challenge with West Ham's Nikola Vlasic which saw the Clarets midfielder call for immediate medical attention.
The game was stopped for around 10 minutes as Westwood received treatment, with Vlasic breaking down in tears following the innocuous challenge in the middle of the pitch.
Three minutes after Westwood had been stretchered off to warm applause, and after a medical emergency in the crowd, Burnley took the lead.
Weghorst scored just his second goal for the club, heading in the rebound after Jay Rodriguez had headed Cornet's corner against the bar.
Cornet then missed the chance to double the visitors' lead. After being released one-on-one by substitute Josh Brownhill, who had replaced Westwood, he was brought down by the onrushing Fabianksi.
The Ivorian took the spot-kick but, having sent Fabianski the wrong way, the winger placed his shot wide of the right post.
His miss was punished in the 74th minute by Soucek, who bundled in from Lanzini's free-kick to score his sixth goal of the season.
Cornet had Pope to thank to ensure his penalty miss did not become even more costly as the England goalkeeper first denied Diop with a fantastic one-handed save from the defender's downward header in the 88th minute.
Then, a minute later, Pope was called into action again as he stopped Antonio's one-on-one attempt to earn Burnley a point.
Opta stats: West Ham's post-Europa League struggles
- West Ham have only won three of their 10 Premier League games that have immediately followed a Europa League match this season (D2 L5).
- No team has drawn more Premier League matches this season than Burnley (13), with this now the most draws the Clarets have ever had in a single Premier League campaign.
- Burnley became the 20th and final Premier League side to take a penalty this season - however, Maxwel Cornet became only the third Clarets player to fail to convert a penalty in the Premier League.
- Michail Antonio had seven shots without scoring for West Ham today and has failed to score in his last 13 Premier League appearances, having 26 efforts at goal without success since last scoring on New Year's Day.
- Only Spurs (52) have given away more Premier League penalties on home soil than West Ham.
Moyes: We've missed an opportunity
West Ham boss David Moyes:
"It's a game where we've missed an opportunity. We had a chance today to really challenge some of the teams closer to the top but we've given that up today by the result.
"There's a lot of things in the game that altered that. We started well and then there was the terrible injury to Ashley Westwood, which obviously derailed the speed of the game for a while and we just lost concentration during that period.
"But that's our fault and nobody else's fault."
Jackson: It's a terrific result
Burnley stand-in manager Mike Jackson:
"In the end it's a good point for us. The first half we done really well and then we've had that penalty miss; I think that sometimes stays with the group a little bit.
"In the second half we rode our luck a little bit at times but it's a terrific result.
"The incident with Ash affected the group... because he's their mate. We're just hoping he's okay. Given everything that has gone on this week and what has happened in the game today, it's a good point for us, in the end."
Man of the Match - Nick Pope
The Burnley goalkeeper was in fine form as he made three top-class saves to deny West Ham, with all three saves displaying his super shot-stopping ability.
The first save came early on against Bowen as he dived full stretch to tip over the West Ham forward's effort from inside the box with his left hand.
The second, arguably the best of the three, saw the England goalkeeper deny Diop with a spectacular stop as he scooped the defender's effort away with his right hand and then he had to be alert just a minute later to stop Antonio in a one-on-one chance.
Without Pope, Burnley would not have come away with a point.
What's next?
West Ham travel to Chelsea on Super Sunday at 2pm - live on Sky Sports Premier League - while Burnley host Southampton on Thursday at 7.45pm - live on Sky Sports Premier League.