England goalkeeper Jordan Pickford was the star performer for the Toffees as he made two crucial saves to earn his side a point on the opening day.
When Morgan Schneiderlin picked up a second yellow for a foul on Luka Milivojevic, Palace felt that there was a window of opportunity for them to pick up all three points.
However, Pickford, who endured a mixed 2018-19 campaign, blocked two efforts from Jordan Ayew and then Max Meyer, and ensured the visitors do not return home empty-handed.
Marco Silva was full of praise for the former Sunderland goalkeeper after the match.
“It was a very, very good game from Jordan,” Silva said to the Liverpool Echo.
“He did really well and it is everything I want from him – and not just because of those two good saves in a key moment in the match – but that he always has full focus on the pitch, always communicating with his team-mates.
“After this he has the quality and he showed that this afternoon. I have no doubts he will keep growing as a football player.”
Last season, Pickford was heavily criticised for his lack of concentration and poor decision making at times. However, on this occasion, he produced an outstanding display, and made two very good blocks to deny Ayew and Meyer.
Everton managed 65% of possession during the game, but the Toffees failed to win their opening game once again, having drawn their opening game in six of the last seven Premier League campaigns.