When Murray stroked home Brighton's third from the spot 15 minutes from time, it triggered a mass walkout from the home supporters - with some aiming their frustration towards the West Ham owners in the directors box.
The Hammers have kept a clean sheet in five of their last seven Premier League games at the London Stadium.
"The board will do what they are going to do".
Pascal Gross maintained his run of being involved in every one of Brighton's goals this season by whipping in a free-kick which was flicked home by Murray with just 10 minutes on the clock. The Spaniard has been in fine form for West Ham's U23 squad, scoring 23 goals in 26 games.
Manuel Lanzini's shot was deflected behind, Pedro Obiang's drive was held by Brighton's No.1 Matt Ryan and defender Shane Duffy made a courageous block to deny Javier Hernandez.
Victory over Brighton could push West Ham into the top 10 of the Premier League for the first time this season.
However, when the ball eventually reached Izquierdo he cut inside on his right foot and unleashed a fierce curler which Hart got a hand to but could not keep out.
How Marko Arnautovic escaped the hook was anyone's guess, although when the club record signing was finally hauled off midway through the second half he was left in little doubt about what the locals made of his display.
The loss drops West Ham down to 17th, two points above the relegation zone, with the alarm bells ringing for Bilic.