While second-placed Tottenham, four points behind Chelsea with six games left, are the form team in the league with seven consecutive wins, Palace have won six of their last eight to surge clear of the bottom three, including beating leaders Chelsea, Arsenal and Liverpool.
Palace's 2-1 victory at Anfield on Sunday, sealed with two goals by former Liverpool striker Christian Benteke, means the Eagles have 38 points in 12th place and look all but safe.
Whatever the result of Chelsea's home game against Southampton on Tuesday, Tottenham will need three points at Selhurst Park to maintain their push for a first English league title since 1961.
But Palace have been transformed by Allardyce, who lost his job as England manager this season after making inappropriate comments in a newspaper sting but returned with a bang after Palace sacked Alan Pardew in December.
They have managed 19 points in their last eight games compared to the 19 they managed in their previous 25.
"It's our spirit, our mental resilience and our belief now that means even when we go a goal or two goals down we don't think we're going to get beaten, he told the Evening Standard.
"Because of that mental resilience we are fighting back now and getting results, and the more results we get the more confidence we build and the results we've had recently nobody would have expected."
"They'd have expected one shock, maybe, but we've done three shocks in as many weeks. It's Chelsea away, Arsenal at home and now Liverpool away. We had the hardest run-in and still have got a very difficult run-in with Tottenham coming up, Manchester City and Manchester United away and Hull City and Burnley.
"But we're nearly there, we've got 38 points with an outstanding run from the players."
Tottenham will be smarting from their 4-2 FA Cup-semi-final defeat by Chelsea on Saturday.
"We have a massive game on Wednesday," Tottenham defender Eric Dier told the club's website. "We have to get (the Cup defeat) out of our heads and carry on -- I'm sure we will."
(Reporting by Martyn Herman; Editing by Gareth Jones)
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