Palace controlled the majority of the game, particularly the first half, with Jefferson Lerma having a header ruled out and Ismaïla Sarr smashing the ball back off the crossbar from close range after a fine team move.
But it was Everton who claimed all three points. Beto gave them the lead before half-time and despite Jean-Philippe Mateta's equaliser not long after the break - Carlos Alcaraz drilled home what would prove to be the winner ten minutes from full-time.
"We know Everton's strengths," Chilwell told Premier League Productions after the game. "They played to their strengths and got two goals that suited their game.
"I think we did well to get back into the game straight after half-time and we fought well. It wasn't a bad performance. Now we've got a week to prepare for the next game.
"We reacted well (to the first goal) got ourselves back into the game. And then at the end, they just got the late winner. But even after that we fought and tried to push on to get an equaliser. I think the mindset and the fight was there for the whole 90 minutes."
Despite coming out on the losing side on his home debut, Chilwell enjoyed his first taste of Selhurst Park and is determined to help the Eagles push on.
"I've come in and tried to help the team today and will continue to help the team in the changing room and on the pitch.
"I tried to do that when I played today and hopefully - whether it's home or away - we can try and pick up wins and climb up the table."