With five of Palace's eight remaining games against teams in the current top six, Allardyce said his side had the toughest run of fixtures and that their recent winning streak had kept their hopes of staying up alive.
"Who can hold their nerve usually decides staying up. You're into single figure games, the bell starts ringing. Time to stand up," Allardyce told a news conference.
"It's not 'arguably' we've got the hardest run-in, we 'have' got the hardest run-in. If we hadn't done what we've done (win four games in a row), we'd be dead and buried now. This is tough, particularly mentally. Pressure builds."
An untimely injury crisis has compounded Allardyce's problems, with at least seven players out of Monday's home match against Arsenal and midfielder James McArthur also a doubt with a back problem.
Palace have three games in six days at the end of the month and Allardyce is desperate for his injured players, including Patrick van Aanholt, James Tomkins, Pape Souare, Yohan Cabaye, Connor Wickham, Loic Remy and Fraizer Campbell, to recover.
"It's going to be tough that week for us. Six days, three games. I need those injury boys fit for that week," added the 62-year-old, who has never been relegated from England's top flight.
(Reporting by Shravanth Vijayakumar in Bengaluru; editing by Mark Heinrich)
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