In a statement, the club said it had launched a "full and thorough investigation" and is "committed to taking decisive and appropriate action".
Ashley Barnes fired Burnley ahead in the 66th minute, prompting one fan to run on to the pitch where he grappled with West Ham skipper Mark Noble.
Once more the stewards were seemingly powerless to intervene except to ask West Ham joint-owners David Gold and David Sullivan, whose decision to quit the east London club's longstanding Upton Park ground for a move to what was the 2012 Olympic Stadium is a source of some fans' fury, to leave their seats, seemingly for their own safety.
The Premier League said: "It is essential that everybody who plays or attends a Premier League football match can do so safely".
"Everyone knows you don't come on the pitch - that's the rules".
"We had played well enough to be in front but the first goal quickly altered things".
He said he hopes the fans got their anger off their chest for future home matches, and admitted playing is a struggle right now.
There is no place at any level of the game for what happened at the London Stadium today.
"Because I think if you look at the majority of the teams around them, I think West Brom - and probably West Brom supporters as well have probably written them off to a certain extent - I think Stoke's supporters a lot of them are probably looking at it and think it's going to be really tough. There will be no further comment at this time". Having tried to play a 3-4-3 formation in recent games, Moyes is expected to stick with this tactic but should see Antonio return to the side and may prefer to play Marko Arnautović at the tip of attack to provide more presence than Javier Hernández who finds himself too isolated in this formation.
Their success and that of their respective teams could be determined by the form of the players around them. We need to play better for longer periods in the game. They have ways of venting that [frustration].