Hodgson made one change to the team that beat Manchester United last week, with Eberechi Eze replacing Jeffrey Schlupp in the starting line up.
Everton took the lead early through Dominic Calvert-Lewin, but Palace equalised via a Cheikhou Kouyate header in the first half.
However, a controversial VAR penalty for handball by Joel Ward afforded Richarlison the opportunity to put the visitors back in front.
It proved to be the last goal of the game, as Palace battled on without reward in the second half.
Follow everything Hodgson has to say about the match below.