An impressive run in April saw Palace climb to 12th in the table but three consecutive defeats, including last Saturday's 5-0 loss to Manchester City has left them at 16th, four points above 18th-placed Hull, with two games left to play.
Three successive defeats, culminating in this humbling at the hands of a Manchester City side showcasing the ruthlessness too often missing during an underwhelming first season under Pep Guardiola, have increased the pressure and Allardyce has called for an urgent refocusing against Hull.
"We are the best team in the league when it comes to creating chances - there is no competition".
"There are four incredible teams fighting for the last two Champions League places", the Catalan boss said.
Guardiola said: "I am pretty sure (former manager) Manuel Pellegrini missed him a lot last season".
Pep Guardiola revealed that he wants Manchester United to win the Europa League and finish fifth, so that England can have five teams in the Champions League. "We have to be man enough and capable enough to do it ourselves, especially after what we've achieved recently", added Allardyce.
The Argentinian, who joined in 2014, has a chance to impress in the remaining games of the season with Bravo suffering from a calf injury. The most hard thing in football, basketball, tennis, in all the sports is to do it in the last moment.
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More chances followed and third goal City had been threatening arrived shortly before the hour mark when the influential De Bruyne steered a shot under Hennessey and just inside the post after Sterling had held off Schlupp to reach Silva's header and set up the chance.
With City leading 1-0 at half-time, Kompany kick-started a second-half goal rush when he hammered an effort into the top-right corner from the centre of the box in the 49th minute. It's not about always playing well or badly, it's not about running more or less, it's about when you must win you win.
De Bruyne and Raheem Sterling were the next to get on the scoresheet before Nicolas Otamendi headed home the Belgian's free-kick in stoppage time.
Palace were in danger of being overrun and the fleet-footed Leroy Sane forced Hennessey into two good saves.
A win against already relegated Sunderland would have moved them to within a point of 16th place Crystal Palace and would have handed them the momentum heading into the final two games of the season.
There are four unbelievable teams fighting for the final two places. "We will see if we can make a step forward and qualify. It will be fantastic", Guardiola added.
Meanwhile, Crystal Palace might have to deal with the absence of Andros Townsend, who is an injury doubt after suffering what manager Sam Allardyce described as a "twinge" in his Achilles during training on Friday. "We managed to bring (the gap) back to four points but after today, it's almost impossible, but we will give it everything for the last three games".
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