The Hammers shrugged off their recent home misery by blitzing Gus Poyet's play-off hopefuls with three goals in the first 10 minutes, before finishing them off in the second half.
"That was overdue and that is the problem," admitted the Upton Park boss, adding: "We're still feeling the pressure as we have no margin for error. It's got to be three wins, starting on Tuesday, and then we can rush and see if anyone has slipped up."
Portuguese winger Ricardo Vaz Te started the rout with a quickfire brace before skipper Kevin Nolan added the third. Vaz Te struck again with a superb scissor kick just after the hour mark to complete a sparkling hat-trick and take his tally to five goals in his last two matches.
Henri Lansbury deflected Carlton Cole's shot in for number five and Brighton's misery was complete when Gary Dicker put through his own net.
It means Allardyce's side, who had slipped off the pace following a rotten run of seven home matches without a win, are now just three points behind second-placed Southampton with three games to play.
"We needed three points and to win them in such style was thrilling and gives everyone a little bit more belief we can do what we wanted to do at the start of the season. Everything about the game was fantastic from start to finish and they were a well-deserved six goals."
Brighton manager Gus Poyet has maintained for most of the season his team are not ready for promotion, and they certainly proved him right.
The defeat had echoes of their 6-1 FA Cup drubbing at Liverpool earlier this season, in which they scored three own-goals.
"We knew how West Ham would start but we didn't deal with it," said Poyet. "The result is embarrassing. I said a few months ago we were not ready and I think Liverpool and West Ham have proved it."