Dwight McNeil struck twice for the Toffees after Marc Guehi had put Palace ahead.
Dyche said afterwards: "I’m delighted for the players, they worked very hard to change a scoreline around.
"You know there’s been question marks over many things and of course there’s been a bit of relief there for me today and relief for the players and that’s a human thing.
"There’s a lot of demand here that we put on ourselves and we have to answer to that and I thought we did today. We certainly did enough to win the game and that’s what was important."
On McNeil, he also said: "It's great to see him back at that level again after that bad incident when he was at Wolves. It was tough for him to come back.
"In an important moment of his career, when he was struggling for goals, we showed the trust in him, and believed in him. He got used to scoring before the incident but we trusted him. From the first moment in this club he showed his hard work and his quality. It's nice to see him score and be decisive for us."