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.
5. Ben Mee 5.4
Dealt with everything that West Ham put into the box from the wide areas, using any limb he could to cut out crosses, but more vulnerable when the Hammers had the ball in front of him. The visitors fashioned a number of shooting opportunities inside the box.
6. Josh Brownhill 5.9
Might not have been afforded the openings to swing the ball into the box, but linked play well and manufactured space for Lowton to deliver from deep. Put in the hard yards and tested Fabianski with a fine effort in the second half.
7. Dwight McNeil 4.9
Struggled to make an impact. Crowded out when on the ball and unable to deliver the ball into the box. Fizzled out as the game wore on and replaced. Might have been able to get closer to Coufal to prevent the cross for the equaliser, but was sucked in.
8. Ashley Westwood 5
Lovely pass over the top to send Wood on his way for the opener. Continued to try and play Burnley in behind, but his approach grew predictable. Deliveries from set-pieces were good, but found it difficult to contend with West Ham in the middle of the park.