Tottenham face West Ham next week.
Jose Izquierdo got the visitors' other goal as the promoted Seagulls jumped up to 10th.
West Ham briefly threatened to make a game of it but Brighton showed the hosts how to defend.
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. He promptly made it seven in the 10th minute when he floated a free-kick towards the head of Murray.
"They threw a lot at us - a lot of crosses and blocked shots".
Brighton, sensationally, doubled their lead in first-half stoppage time.
Brighton were then awarded a penalty in the 74th minute when Murray was brought down by Pablo Zabaleta. "I think we deserved it".
In Chris Hughton's 100th game as a Premier League manager, the Seagulls sensed nervousness in West Ham's play and struck an unlikely opener.
Hammers joint-owner David Sullivan had backed his manager before kick-off, saying he would be given the opportunity to at least see out his contract until the end of the season, but the Croatian's future now seems less secure. "All I can say is that it was very disappointing and frustrating and as manager I take full responsibility for that", he told the club's website (www.whufc.com).
The 37-year-old Bruno said: "To play away from home always is tough in Premier League, to get three points is so important for us".
"I don't want to hide".