Burnley's record of never having won a Premier League game in the capital continued on their first visit to London Stadium.
But the Clarets will wonder how they left West Ham's new ground with nothing to show for their endeavours after an Olympian effort to stop the rot on the road.
The bare facts show that Sean Dyche's men are still to win - or score from open play - away from home this season as West Ham eased some of the pressure on their position near the drop-zone with only a third league win at the former home of the London Games.
But this was by far the Clarets' most positive performance away from their Turf Moor stronghold as they recovered from a slow start and a controversial goal in first-half stoppage time, scored by Mark Noble after his penalty had been saved by Tom Heaton, to dominate the second half.
This time the chances came to put at least another point on the board, but Scott Arfield was twice denied by goalkeeper Darren Randolph and Sam Vokes couldn't apply the finishing touches to a battling personal performance as the Hammers somehow held out.
Dyche paired Vokes and Andre Gray together in attack for the first time since the Clarets' 0-0 draw at Manchester United at the end of October - the only time Burnley had previously taken an away point this season.
Gray's re-call came in place of midfielder Steven Defour as Dyche reverted to a 4-4-2 line-up, the same formation he deployed at Stoke just over a week ago where the Clarets pleased their boss with the performance, if not the result.
But the early pressure came from the Hammers with their returning striker, Andy Carroll, heading wide a cross from Michail Antonio inside the first two minutes.
Play-maker Dimitri Payet then had a shot comfortably gathered by goalkeeper Heaton and Manuel Lanzini shot over from long range.
The muscular Carroll threatened again after holding off Ben Mee to meet a lofted Payet pass with a shot he also struck over the top.
But Burnley did what they needed to do in toughing it out to get a foothold in the game and, as Andre Gray and Jeff Hendrick had shots charged down, begin to frustrate a home crowd which is yet to enjoy life in their new home.
The Hammers replied with a skidding effort from Antonio which was easily stopped by Heaton but the Clarets' keeper was at full stretch 10 minutes before the break when Perdo Obiang moved forward to hit a fierce drive against the outside of a post.
The home side hit the woodwork again through Noble's curling shot but just when it looked as though the Clarets would get to half-time on level terms thanks to committed defending - epitomised by Matt Lowton's superb block from Aaron Cresswell - they conceded a controversial penalty in stoppage time.
Heaton was prevented from getting a clean punch to another Carroll header by the raised arm of Cheikhou Kouyate - which may also have made contact with the ball - following a Lanzini corner and when Winston Reid shaped to shoot, the defender was clipped from behind by Mee.
The Clarets' captain saved Noble's spot-kick, diving to his left, but couldn't hold onto the ball as his opposite number converted the rebound to make it 1-0.
With their away-day goal drought heading into a seventh game, the Clarets needed to break new ground to get any reward and extend the Hammers' six-match winless run.
And the visitors started the second half in positive style as they took the initiative and came close to forcing a leveler three times in the opening 15 minutes.
The first two chances came from free-kicks with the Hammers pinned in their box, as Vokes flicked an overhead effort just wide from Michael Keane's header and then just failed to get a decisive touch to Mee's knock-down.
The tireless Vokes was knocked from pillar to post in a bruising encounter but the goal his efforts deserved continued to elude him, with his next opportunity with an hour gone the best yet.
The Clarets' slickest move of the night opened up the Hammers' defence and after Arfield was superbly denied by Randolph, Vokes was unable to direct the bouncing ball goalwards as Reid did just enough to force the Clarets' striker to head over.
It was all Burnley, though, and with 18 minutes to go they were only inches away again from making it 1-1.
Arfield came alive in the second half and when given a shooting chance from a free-kick 25 yards out, the midfielder looked to have beaten Randolph with a superb strike until the Republic of Ireland international got across at the last moment to make a fine save.
West Ham had barely threatened since going in front but with the Clarets unable to conjur up one more chance to take a deserved point, they almost grabbed a second goal in injury time as substitute Andre Ayew shot inches wide as the relieved home supporters waited to hear the final whistle.
Burnley: Heaton, Lowton, Keane, Mee, Ward, Boyd (Barnes 76), Marney, Hendrick, Arfield (Bamford 90+4), Vokes, Gray. Substitutes: Robinson, Flanagan, Kightly, Defour, Tarkowski.
West Ham: Randolph, Antonio, Ogbonna, Reid, Cresswell, Obiang, Noble (Nordtveit 89), Kouyate, Lanzini (Ayew 69), Payet, Carroll (Fletcher 82). Substitutes: Adrian, Fernandes, Quina, Browne.
Referee: Bobby Madley