Argentine forward Zarate joined the Hammers from Velez Sarsfield in pre-season and he marked his maiden appearance in the Premier League with a cool finish in the first half at Selhurst Park.
Stewart Downing netted a fine second for the Hammers moments later before Marouane Chamakh reduced the deficit early in the second half.
Carlton Cole's goal sealed West Ham's first win of the season following their last-gasp loss to Tottenham last weekend and eased the pressure on boss Sam Allardyce.
Palace's second successive defeat plunged the south London club into further turmoil, with co-owner Steve Parish still hunting a replacement for boss Tony Pulis, who quit on the eve of the new season after a row over transfer policy.
Parish thought he had lined up Pulis's successor earlier this week when he offered the job to Malky Mackay.
But he had to rescind that offer following the newspaper revelations about inappropriate text messages sent between Mackay and Iain Moody when the pair worked together at Cardiff.
The fall-out from that scandal also led Moody to quit as Palace's sporting director in midweek and the end of what Parish described as a "difficult week" was just as traumatic.
Former Tottenham manager Tim Sherwood pulled out of the running for the Palace job after initially being deemed second choice to Mackay, leaving caretaker boss Keith Millen to take charge for the second week running.
Millen saw former Lazio striker Zarate cap a lively debut with an exquisite strike to open the scoring in the 34th minute.
Hammers boss Allardyce has only clung onto his job after promising to play more expansive, attacking football and this enterprising display was a step in the right direction.
Downing claimed just his second West Ham goal three minutes later as Palace paid the price for backing off in midfield.
He raced down the right, cut off his flank and nudged a side-footed effort past goalkeeper Julian Speroni.
James Tomkins headed against the crossbar to open the second half as West Ham sought to finish off the contest.
Palace immediately seized on the reprieve, former Arsenal forward Chamakh firing in a low shot from 20 yards to half the deficit in the 48th minute.
But Cole, so profligate against Tottenham, pounced on a loose ball in the Palace area to lash past Speroni and kill the contest in the 62nd minute.
Parish must now act swiftly to secure a new managerial appointment, with Palace battling to get their season back on track.