Goals from Jason Puncheon, Scott Dann and Cameron Jerome secured all three points for Crystal Palace at Goodison Park to take the Eagles to the magic 40-point survival mark
The Eagles were six points adrift of safety when the former Stoke City boss was appointed and are now 10 points clear of the drop-zone with four games left to play and the Welshman paid tribute to his squad for their part in their remarkable turnaround since his arrival.
"I'm absolutely delighted for the supporters, for the football club," Pulis told reporters. "It's nice to say that we are going to be a Premier League club with four games to go. It's remarkable, really.
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Pulis revealed that he had been unsure about taking the position at Selhurst Park in November but that some words of encouragement from Manchester United legend Sir Alex Ferguson made up his mind.
"Everybody I spoke to said it is a difficult job," he recalled. "Alex Ferguson and my old chairman [at Stoke], Peter Coates, were the only two who said to take the challenge."
Meanwhile, Toffees boss Roberto Martinez refused to accept defeat in the race for Champions League football despite ceding the initiative to fourth-placed Arsenal.
He argued: "I don't think Arsenal will get full points from now until the end of the season. When you are one point away from them, with the goal difference in our favour, it is going to be very much relying on what we do.
"I know that you look at the table and think that it is out of our hands but I truly believe that it is in our hands and it is down to how many points we can get."