Glenn Murray scored his first Premier League goal in almost two years and Izquierdo hit his first for the club as West Ham fans turned on their manager.
Right from the start, the Brighton fans made the most noise, and so did their team when Murray headed them in front.
Hammers boss Bilic decided to recall Pedro Obiang to the starting line-up but his decision backfired with just ten minutes on the clock.
First the midfielder gave away a cheap free kick in a dangerous area, then he failed to mark Murray from the resulting cross by Pascal Gross and just watched as he nodded it in.
Murray scored 23 times during Brighton’s promotion season but that was his first goal since May, and he will rarely score an easier one.
West Ham winger Michail Antonio then survived an injury scare after jarring his knee in an awkward landing.
When the Hammers rallied, they peppered the Brighton goal with shots but the Seagulls weathered the storm, putting their bodies on the line for the cause.
Lewis Dunk and Shane Duffy both threw themselves into blocks but it was all too easy with West Ham missing suspended striker Andy Carroll badly.
All the while Brighton continued to look dangerous, with Joe Hart forced into a decent save from Murray.
But Hart couldn’t stop Izquierdo’s shot right at the end of the half, despite getting a hand to it, and when the half-time whistle went the boos rained down on Bilic and co.
Pressure back on then. Co-owner Davild Sullivan, who has not seen eye to eye with his manager on a range of issues, made his thoughts clear even before kick-off.
Writing in the programme, he insisted Bilic had “a wealth of options available to him” even without Carroll, and called this a “huge” game he expected to win.
Given the fact Bilic played holding midfielder cum defender Cheikhou Kouyate up front at times in the first half, and then took him off, he might beg to differ.
If Hammers fans were expecting an immediate response from their side they were sadly mistaken, and you could feel the frustration grow.
You couldn’t really blame them. Record signing Marko Arnautovic strolled around lazily. Sparkplug Manuel Lanzini was anonymous. Jose Fonte trod water.
At times, Brighton looked so comfortable they could have been settling into an armchair with a nice mug of hot tea and a plate of biscuits.
So it didn’t take long for the boos and whistles to start up again, with Bilic, deep in conversation with his coaching staff, seemingly at a loss as to how to change it.
Javier Hernandez, signed to score goals, found himself having to drop deeper and deeper in search of the ball, and when he finally received it, he had few options ahead of him.
Lanzini had a free kick saved but a fightback never really looked like it was on the cards for the Hammers.
Murray finished the game off after Pablo Zabaleta gave away a penalty late on, sparking a mass walkout from Hammer fans.
