The Hammers fell to a third consecutive defeat as Sean Dyche's charges scored three unanswered second-half goals at the London Stadium.
Each goal was followed up by incursions onto the pitch by home supporters, with one individual approaching Hammers midfielder Mark Noble after Burnley's opener.
"Up until the goal you wouldn't have thought that it would have been such a bad day," West Ham boss Moyes told BBC Sport. "We had played well enough to be in front but the first goal quickly altered things.
"We want the supporters behind us, since I've been here they've been really good. But you can't cross the line and come on the pitch. A lot of my players did well with the way they acted.
"We still have plenty of games, we are good enough and we will be out of the bottom three come the end of the season. I'm confident we have the players here."
The result puts West Ham in 16th place in the table, three points from the bottom three.