Substitute Chris Wood scored twice after Ashley Barnes's 66th-minute opener as Sean Dyche's visitors made it back-to-back wins in the Premier League.
However, the focus was less about football and more on the actions of West Ham's fans, with three separate pitch invasions forcing referee Lee Mason to hold up proceedings.
A pair of fans entered the field of play - with one clashing with West Ham captain Mark Noble - after Barnes fired a left-footed rocket beyond the recalled Joe Hart to break the deadlock.
There were also further stoppages in play after both of Wood's goals, with one intruder planting the corner flag in the centre circle in a show of frustration.
Others vented their feelings about co-owners David Sullivan and David Gold in the stands as a day when the club paid tribute to former captain Bobby Moore - who died 25 years ago - became memorable for all the wrong reasons.
Sullivan and Gold were forced to evacuate their seats while a large number of the home crowd remained after the final whistle to continue their complaints.
West Ham's problems are not just confined to an unhappy fanbase, though. A third successive Premier League loss slips them down to 16th in the table, just three points above the relegation zone.
Hart returned between the posts as David Moyes made five changes to his starting XI after back-to-back 4-1 defeats, yet it was Burnley's Nick Pope – the 25-year-old who has capitalised on Tom Heaton's unfortunate shoulder injury to emerge as a contender to make England's 2018 FIFA World Cup squad – who was the busier of the two keepers in a scoreless first half.
The visitors relied on their last line of defence to deny Marko Arnautovic and Manuel Lanzini, Pope using his feet to good effect to turn away low drives.
West Ham felt aggrieved not to be awarded a penalty before the break when Aaron Cresswell's cross struck Aaron Lennon's arm, with referee Mason dismissing the vociferous appeals.
Burnley failed to have a shot on target in the opening half, but thought they had taken the lead when Barnes fired low underneath Hart in the 53rd minute.
