The Eagles had struck the crossbar in the first half through Cheikhou Kouyaté before winning a spot-kick just before the hour mark, but Jordan Pickford denied the visiting skipper with his legs.
However with three minutes remaining it still looked like Roy Hodgson’s team were set for a deserved point, but the two Toffees substitutes struck in the 87th and 89th minutes to stretch Everton’s unbeaten run against Palace to eight matches.
Marco Silva’s team started impressively with some slick, quick passing allowing them to spend plenty of time inside the final third, but a well-drilled Palace defence repelled everything their opponents threw at them except a Seamus Coleman blast which harmlessly flew over on nine minutes.
The Eagles continued to keep Everton at bay until the 30th minute when perennial scourge Gylfi Sigurdsson played a one-two with Coleman on the edge of the box before stabbing the ball towards the top corner which the in-form Wayne Hennessey managed to divert over, but Hodgson’s team showed what they could create themselves as they went close three times in quick succession.
Firstly, Andros Townsend clipped a cross towards the back post which James Tomkins headed well wide on 33 minutes, and 60 seconds later Milivojevic forced Jordan Pickford to scramble across his goalline to push away a curling free-kick. Then, from the resulting corner, they almost edged in front when Tomkins nodded the delivery across goal and Kouyaté headed the ball off the crossbar with England’s number one well beaten.
A quiet affair had suddenly burst into life and Richarlison was next to test his luck with a header from a corner which whistled over, and just before the half-time whistle Sigurdsson came closer with a drive from 20 yards which zipped past the bottom corner.
In similar fashion to the first half, Everton enjoyed the early stages without creating anything dangerous, but just before the 60-minute mark Palace were handed a huge chance to go in front. For the first time all game, Wilfried Zaha teased his way into the box and cut back, deceiving Coleman who brought him down for a penalty. Milivojevic attempted to drill it down the middle, but the diving Pickford dangled out a trailing leg and managed to prevent the kick finding the net.
Once again that sparked proceedings back into life and at the other end Hennessey showed his own importance once again when he denied Theo Walcott when he went one-on-one with Wales’ number one, and Townsend tried to bend the ball into the top corner which Pickford was equal to.
It was anyone’s game by this point and on 74 minutes Andre Gomes dragged an effort wide following a swift break by Walcott before Sigurdsson saw a shot from a similar range blocked by the resolute Eagles defence
But with three minutes remaining, Silva’s men got the golden goal. Michael Keane won a header in the middle of the pitch which fell to Richarlison and he played it to Ademola Lookman. The former Charlton Athletic man scooped it into the box where the fresh Calvert-Lewin was waiting to pounce, and he headed past a helpless Hennessey from close range just four minutes after his introduction.
Then, two minutes later Keane lumped a long ball forward and another substitute Tosun was able to outsprint Mamadou Sakho from the halfway line to seize possession, compose himself and fire through Hennessey’s legs to seal a third straight win for the Merseysiders.
Everton: Pickford, Coleman, Keane, Zouma, Digne, Gomes (Calvert-Lewin 82), Gueye, Walcott (Lookman 82), Sigurdsson, Bernard (Tosun 66), Richarlison. Subs not used: Stekelenburg, Baines, Mina, Davies.
Palace: Hennessey, Wan-Bissaka, Tomkins, Sakho, Van Aanholt, McArthur (Wickham 89), Milivojevic, Kouyaté, Schlupp, Townsend, Zaha. Subs not used: Guaita, Kelly, Riedewald, Puncheon, Kaikai, Ayew.