Palace started the second-half strongly, enjoying more of the ball and playing further up the pitch. On the hour-mark, Ademola Ola-Adebomi saw a header saved by Leban and a follow up effort from skipper Jack Wells-Morrison went just wide of the post.
Raymond was again at the forefront of Palace attacks, as he fired wide just a few minutes later. He managed to get on the end of a driven ball from Adaramola, but couldn’t direct his effort goalwards.
Everton’s best chance of the second-half came from Liam Higgins cross-cum-shot which was driven at Owen Goodman’s near post. The Palace ‘keeper managed to react brilliantly to get a fingertip to the effort and turn it behind for a corner.
As the game headed in to the final 10 minutes, it looked as though the points were going to be shared, however second-half substitute Victor Akinwale had other ideas.
Following a corner, Wells-Morrison and Ola-Adebomi combined well to keep the ball alive in the second-phase of play - the latter played in Rodney, who stood a great ball up to the six-yard box, to find Akinwale rising highest to head in his fifth league goal of the season.
With Palace a goal to the good, they looked to see out the rest of the game. A second was almost secured by substitute Kofi Balmer, who returned from a calf problem, though his header from a late corner was smothered by the Leban.
After four minutes of time added on, the referee brought an end to proceedings. The victory over the Toffees sees McCarthy’s side move up to third place, two points clear of Fulham and four behind Chelsea.
Everton: Leban (GK), Patterson, Samuels-Smith, Welch, Campbell, Metcalfe (Kouyate, 55), Higgins, Price, Okornkwo, Heath (Butterfield, 55), McAllister.
Subs not used: Barrett (GK), John, Bates.
Crystal Palace: Goodman (GK), Imray (Rodney, 34), Grehan, Ferguson (Balmer, 62), Watson, Adaramola, Ozoh, Wells-Morrison, Raymond, Omilabu (Akinwale, 62), Ola-Adebomi.
Subs not used: Izquierdo (GK), Cadogan.