Palace were soon back in the ascendancy and by the 24th minute, the Londoners had netted their third of the match as Zaha beat his marker at the back post to head into the bottom corner from Milivojevic's pinpoint cross from a central position.
To the outsider, Crystal Palace against Brighton, some 43 miles apart, may appear to one of football's more contrived derbies.
Crystal Palace find themselves one place above the relegation zone with five games left.
The central characters were Terry Venables, whose appointment at Selhurst Park in 1976 was quickly followed by his former Tottenham room-mate Alan Mullery taking charge of Brighton.
It is, though, hard to see tomorrow passing off so quietly.
In added-on time, Murray squandered a chance from close range after the ball had been flicked into his path, and it proved to be Brighton's final opening as Palace held on for what could prove to be a crucial three points come the end of the season.
Fourth bottom Palace now boast a precarious three-point advantage from the relegation places, but have played a game more than Southampton, who are directly below them.
Salah hits 40 goal mark as Liverpool beat Bournemouth in Premier League
Liverpool reached 300 goals under manager Jurgen Klopp for victory over Bournemouth . It's all about him and all about us, how we finish the season.
"I thought we were good for periods of time".
The talismanic Zaha and Andros Townsend excelled as a front two and Hodgson's verdict was not a glowing endorsement of regular frontman Christian Benteke, absent with a groin injury. They all have their players that can have a big influence, but he's really developed as a player in the last few seasons. Given their subsequent problems - through fitness and form - with various strikers, it must be galling for Palace they allowed Murray to leave for just £5million two-and-a-half years ago.
Saturday's Premier League games produced some massive results at the top and bottom of the top-flight table, including big wins for leaders Manchester City and struggling Crystal Palace.
Brighton completed the scoring in the first half, as winger Jose Izquierdo beat Joel Ward before finishing past Wayne Hennessey in the home goal.
Swansea moved five points ahead of Southampton after a 1-1 home draw with Everton.
But with local bragging rights on the line, as well as precious points, Palace boss Hodgson knows the pressure that his players will be under after previously steering West Brom and Fulham to survival. The events use footballers past and present to help highlight issues about racism in social society.