Hammers captain Mark Noble tried to send one pitch invader away and another sprinted on holding a corner flag shortly after Chris Wood added to Ashley Barnes’ opener for the visitors.
Substitute Wood made it three for Sean Dyche’s men, who narrowed the gap on sixth-placed Arsenal to two points and won their first away game since November.
West Ham’s owners David Sullivan and David Gold were forced to leave the directors’ box near the end as anger boiled over and supporters chanted’sack the board’.
David Moyes’ West Ham are in 16th place, three points from the bottom three.
According to BBC, the home side’s collapse was at odds with an impressive first half when they started well, going close through striker Marko Arnautovic and playmaker Manuel Lanzini, who were denied by the legs of Burnley keeper Nick Pope.
They might have had a first-half penalty too when Aaron Cresswell’s cross struck Burnley winger Aaron Lennon’s arm, but the second-half introduction of New Zealand international Wood for the visitors turned the game as it did during the Clarets’ last win against Everton, when he scored the winner