The high-flying Hammers seemed to be a little lethargic in their endeavours while the hosts looked a little lost without the presence of star-man Maxwel Cornet on the field.
Story of the match
It was a cagey affair with neither goalkeeper really being tested in opening half of the first-half.
The Clarets had the better of the first 10 minutes or so and Chris Wood might have done better with a header from a corner but he couldn't keep his effort down.
The Burnley goalkeeper found himself soon after down in a heap after colliding with Craig Dawson after another West Ham set-piece. The medics were on hand to quickly patch him up and play was under way after a few minutes of treatment.
Dawson was in the wars and he appealed for a penalty after Dwight McNeil appeared to clip the defender's leg. VAR checked the incident and was satisfied with the on-field decision.
In fairness, after an insomnia curing first-half, both teams came out of the traps with a greater attacking desire.
Rodriguez connected onto Charlie Taylor's low cross but Dawson was on hand to make a crucial block.
Rice was starting to take the game by the scruff of the beck and his deflected cross was almost turned in at the back-post by Said Benrahma; the Algerian's header was brilliantly palmed away by Pope.
Sadly for the purist, the last half-an-hour seemed to fizzle out and both teams found it difficult to make those killer passes in the final-third.
However, in the 89th minute, Rodriguez could have won it with a cute flick from Taylor's drilled cross. The ball skidded agonisingly wide of the far post.
All in all, it wasn't one that will live long in the memory. Both teams could have done with a victory but it could still be a big point for the two clubs.