Two late goals in two minutes from substitutes Dominic Calvert-Lewin and Cenc Tosun saw Everton past Crystal Palace -- for whom Luka Milivojevic had a penalty saved -- 2-0 at Goodison Park.
Calvert-Lewin converted a cross by fellow sub Ademola Lookman in the 87th minute with a drab game appearing certain to end in a 0-0 draw before Tosun rounded off a quick break with a low finish.
Milivojevic had opted to put his penalty down the middle but saw it balloon to safety off Jordan Pickford's trailing leg as Palace's best opportunity came and went.
Everton's Richarlison wasted an early chance to shoot, instead trying to backheel to Lucas Digne, but that was a rare moment of intent in a scrappy opening.
Seamus Coleman fired well over from distance before, as the quarter of an hour approached, Richarlison, racing towards the area, ran into defenders when he had passes available.
After 20 minutes, the home side were almost in trouble as Kurt Zouma got a pass inside his own area wrong, forcing Coleman into a hurried clearance as Palace looked to take advantage.
But 10 minutes later, Everton fashioned a chance with a neat move as Coleman combined with Gylfi Sigurdsson, whose shot was pushed away by keeper Wayne Hennessey.
A Milivojevic free kick forced Pickford into a save before, with 36 minutes gone, the bar denied Palace as Cheikhou Kouyate's header following a corner hit the woodwork.
Richarlison headed over as a tepid first half came to a close, but Everton began the second period with more intensity and Milivojevic was booked for fouling the same player.
Palace missed their golden chance to take the lead on the hour when Coleman fouled Wilfried Zaha inside the area, but Milivojevic's penalty was stopped by England keeper Pickford's trailing leg.
Within a couple of minutes, Palace's Hennessey made a good save of his own, thwarting Theo Walcott when the Everton attacker raced into the area and shot.
After 66 minutes Marco Silva made a change, introducing Tosun for Bernard, but Everton were struggling for momentum and Palace threatened when Patrick Van Aanholt and Jeffrey Schlupp combined before the home side cleared their lines.
With a quarter of an hour left, Andre Gomes, making his Everton debut, curled an effort wide.
Coleman was booked for a foul on Zaha before Everton made a double change, Lookman on for Walcott and Calvert-Lewin replacing Gomes -- and the switch proved inspired as the Merseysiders took the points.