The 2-1 win moves Everton further away from the bottom three.
Moyes said, "It was a great win because for large parts we weren't at our best. I think Wednesday night took so much out of them but fortunately we got a goal in front which gave us something.
"I thought our defenders the second half and especially late on, did a great job defensively. I was annoyed with how we played in the first half and then we gave a goal away soon after the start of he second half. I though Crystal Palace were on top and thankfully we got the goal and the three points.
"To come back to Everton was something I hoped would happen and I didn't expect it to happen. It is great to be accepted again and the fans have been brilliant. Even today, we had a brilliant support with us because I think they are starting to believe a bit more in us and the team. Everton has had a difficult period and I certainly think there is light at the end of the tunnel now with the new owners coming in.
"We want the players to be positive and play in the way they feel good about themselves and we want to create chances."