Goals from the in-form Jason Puncheon and defender Scott Dann put Palace into a deserved 2-0 lead at Goodison Park on Wednesday, before Everton fought back with a Steven Naismith goal.
A third from Cameron Jerome sealed the points for the Londoners near the end, meaning that Kevin Mirallas' late strike for Everton was scant consolation.
As well as blowing the race for the final UEFA Champions League spot wide open between Arsenal and Everton, the result also lifted Palace to 11th in the Premier League table with four games to go - leaving them 10 points above the drop zone and all but safe.
But Pulis was quick to share the credit with his squad following a remarkable turnaround since his arrival at the helm in November.
"It is a great result, but I started at Hull (Palace's first game with Pulis in charge) when I think we had four points, and now we have 40 points. So it's a remarkable achievement from the players.
"The players have been absolutely fantastic - I'm proud of them. Without their effort and their commitment and buying in to what we are trying to do, we wouldn't have achieved it.
"It's a different team to what I had at (former club) Stoke, we have pace and I didn't have too much pace at Stoke. We move the ball really quickly through the pitch, and I thought we did that really well today."
