Jean-Philippe Mateta struck for Palace, but Beto and Charly Alcaraz scored the goals for Everton in their 2-1 win.
Beto said, "What a week it has been. It has been a tough week, but two great results. We are happy to have these two great results. We knew it was a tough game today, but we got the three points and we are happy.
"It was a difficult chance as I took a touch, I was calm and composed. I am happy to score and the confidence, last season I wasn't so confident.
"Now I feel all my training and all my mental work and all the work I do with my teammates is paying off.
"Now, I believe. I believe that I belong in the Premier League and I'm a Premier League player."
On manager David Moyes' impact since his return, Beto said: "A lot of work finishing and holding the ball, but apart from that play my game and do my job."
On Mateta, he added: "I like to see his style of play and he is similar to me. He scored once again."