With West Ham set to record a first away win of the season, Michail Antonio, rather than taking the ball to the corner, lost possession and Ruben Loftus-Cheek sped away to set up Zaha for his goal.
Chicharito converted Aaron Cresswell’s cross to open the scoring and Andre Ayew doubled West Ham's lead with a superb solo effort.
Palace were handed a lifeline when Angelo Ogbonna fouled Andros Townsend inside the area and Luka Milivojevic converted the penalty.
The home side chased an equaliser but Hammers goalkeeper Joe Hart denied Jeffrey Schlupp and Yohan Cabaye and then the woodwork thwarted first Cabaye and then substitute Bakary Sako from four yards before Zaha struck.
CRY: 5 Nov v Spurs (A)
WHU: 4 Nov v Liverpool (H)