Palace midfielder Luka Milivojevic missed a penalty on the hour mark, and the Eagles were made to pay as substitutes Dominic Calvert-Lewin and Cenk Tosun netted for Marco Silva's team in the closing stages.
The Toffees are now placed in eighth position in the top-flight standings, while Palace are 15th and sit just two points clear of the relegation zone.
Both teams started well on Merseyside without producing an end product, with Seamus Coleman firing the one half-chance over the crossbar from 20 yards.
Andre Gomes was struggling to get into the game on his Everton debut, but the home side went close just before the half-hour mark when Gylfi Sigurdsson was denied by Wayne Hennessey from 12 yards out.
Cheikhou Kouyate came agonisingly close to putting Palace ahead from the resulting corner, with the midfielder's header hitting the underside of the crossbar before being cleared by the Everton defence.
Kouyate soon sent another headed effort off target, but Everton got back onto the front foot before the break as Sigurdsson curled a shot wide of the far post from outside of the area.
The home side continued to build momentum after the restart but just before the hour mark, Palace were rightly awarded a penalty when Seamus Coleman hooked Wilfried Zaha's foot in the area.
However, the Eagles squandered their opportunity to move in front as Pickford stuck out a leg to prevent Milivojevic's strike - aimed towards the centre of goal - from finding the back of the net.
The home supporters recognised that their team had been given a let-off, but the Toffees soon missed out on taking the lead at the other end when Hennessey kept out Theo Walcott's effort after the winger had been played through on goal.
Andros Townsend saw a speculative attempt for Palace comfortably saved by Pickford, before Gomes curled the ball just wide from 20 yards at the other end.
Marco Silva opted to introduce Ademola Lookman and Calvert-Lewin with eight minutes remaining, and the pair soon combined to put Everton in front with just four minutes remaining.
Lookman received the ball on the left-hand side of the penalty area before managing to find Calvert-Lewin, who was able to power a header past Hennessey from eight yards out.
A second goal arrived within seconds of Palace restarting the game, with a long clearance from Michael Keane releasing Tosun who managed to fire the ball through the legs of Hennessey.
The two goals were harsh on Palace, but the work of Roy Hodgson's men in the final third once again failed to deliver a reward and it is an area of their game which must improve if they are to avoid a relegation scrap.
EVERTON (4-2-3-1) Pickford; Coleman, Keane, Zouma, Digne; Gomes (Calvert-Lewin 82'), Gueye; Walcott (Lookman 82'), Sigurdsson, Bernard (Tosun 67'); Richarlison
CRYSTAL PALACE (4-5-1): Hennessey; Wan-Bissaka, Tomkins, Sakho, Van Aanholt; Townsend, Kouyate, McArthur (Wickham 89'), Milivojevic, Schlupp; Zaha