Sean Dyche' s side went ahead from the spot in the 19th when Chris Wood picked himself up to score for the seventh time against the Hammers in the Premier League.
However, for the third time in five fixtures, the Clarets would go on to surrender their lead and finish the game pointless.
Michail Antonio scored both goals for the visitors, finishing crosses from Vladimir Coufal and Said Benrahma.
Here are the ratings.
Michail Antonio scored both goals for the visitors, finishing crosses from Vladimir Coufal and Said Benrahma.
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West Ham United's Scottish manager David Moyes (L) talks with Burnley's English manager Sean Dyche ahead of the English Premier League football match between West Ham United and Burnley at The London Stadium, in east London on January 16, 2021. Photo: Adam Davy
. Nick Pope 5.8
Left completely exposed by his defence for both Antonio's goals before the break. Commanded his penalty area well, successful with a number of high claims, and saved well from Lingard deep into stoppage time.
. Matt Lowton 4.6
Couldn't live with the pace and trickery of Benrahma, who supplied the cross for West Ham's second. Also culpable for the away side's equaliser, applying little or no pressure on Antonio and continued to be targeted as the Hammers broke forward.
. Charlie Taylor 5.6
Deserved more engagement with his deliveries, particularly in the first half when whipping in several inviting crosses. West Ham's equaliser came down his side, though he had recovered to shut out Lingard's route to goal.
13. Johann Berg Gudmundsson 4.6
Looked much more comfortable on the left hand side, looking to take the game on. Appeared to have the beating of Coufal in the short amount of time he was on the pitch and productive when getting on the ball.