Carlo Ancelotti's side took the lead on the 10th minute when a superb pass by Colombia international James Rodriguez found the overlapping Seamus Coleman who slipped the ball across for Dominic Calvert-Lewin to score for the third straight league game.
But five minutes before the interval, referee Kevin Friend awarded a penalty, ruling that Joel Ward had handled the ball.
Lucas Digne had headed the ball at the Palace defender who appeared to try to pull his arm out of the way but, after being advised to check the pitchside monitor, Friend pointed to the spot and Richarlison drove home the penalty.
"The referee gave the penalty for us and it looked like it hit his hand," Calvert-Lewin said.
"Richarlison put it away. The referees have a tough job whether it is a penalty or not."
Palace dominated after the break but Everton defended well to maintain their 100% record.
While Ancelotti's side could not reproduce much of the attacking flair they showed in wins at Tottenham Hotspur and against West Bromwich Albion, they showed the solidity they will need if they are to push for a European place this season.
"I thought it wasn't our best performance today, but we stuck in there and got the result to keep the momentum going," Calvert-Lewin said.
"It is early days but we are looking to build on every game, the more we are training together, it is exciting times. But we are not getting carried away."