Murray's brace - taking his tally to five in seven meetings with the Hammers - and a delightful effort by Colombian global Jose Izquierdo gave Brighton their first away win of the season and the "Seagulls" first in the top flight since 1983.
West Ham had very little to offer in attack and anything they conjured up was dealt with brilliantly by Brighton central defenders Lewis Dunk and Shane Duffy.
"We're benefiting from the amount of games they've played together and the relationship they have".
West Ham have never overturned a two-goal half-time deficit in the Premier League and - after being booed off by unsatisfied supporters - they did not look like ending that record at the 76th attempt, with Murray wrapping up the win from the penalty spot after Pablo Zabaleta's clumsy challenge in the box.
Tottenham face West Ham next week.
Spurs have already beaten West Ham once this season, winning 3-2 at the London Stadium, and are hoping to inflict further damage on Wednesday night.More news: Ben Simmons Impresses On NBA Debut
Brighton made a bright start and deservedly went in front after 10 minutes when Murray headed a free kick by Pascal Gross into Joe Hart's net.
Then Brighton, having weathered that mild storm, doubled their lead in first-half stoppage time.
Bilic sent Andre Ayew on for Cheikhou Kouyate for the second half but West Ham struggled to break down a Brighton side intent to hold onto their advantage.
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.