The Hammers eased the pressure on the Croatian tactician on Wednesday with a stunning come-from-behind 3-2 win against Tottenham at Wembley in the Carabao Cup.
Ayew scored twice and Angelo Ogbonna's header completed the turnaround after Dele Alli and Moussa Sissoko had put Tottenham in control.
A quarter-final clash at Arsenal awaits, but the Hammers have a big game at Selhurst Park this weekend as they look to put daylight between themselves and the relegation places.
Palace are bottom the table while West Ham are 16th, above the drop zone on goal difference.
"When you're in a club like West Ham and the results are difficult, the manager is under pressure, and the players are too, but we just need to keep working hard," Ayew said.
"The manager knows the players are behind him and will give everything, not just for him but for themselves too because we have to wear the shirt with pride.
"It's a massive win, but we need to stay calm because we've got a massive game on Saturday."