Everton goalkeeper Jordan Pickford saved a penalty before substitutes Dominic Calvert-Lewin and Cenk Tosun scored in the closing minutes as the Merseysiders won 2-0 at home to Crystal Palace on Sunday.
Everton's third successive win in the Premier League lifted them into eighth place, but defeat left Palace just two points above the early season bottom three.
The Eagles, managed by former England boss Roy Hodgson, had a great chance to break the deadlock on the hour mark when Wilfried Zaha was fouled in the box by Seamus Coleman.
Luka Milivojevic hit his penalty down the middle and Pickford, diving left, blocked with his trailing leg.
"I just looked at his run up and he normally whips it, I left my right leg dangling and managed to get a foot on it," Pickford told Sky Sports. "It was a game-changer as it got the crowd going, and we needed that.
Everton manager Marco Silva was then rewarded for his decision to make attacking substitutions with two well-taken goals.
Calvert-Lewin came off the bench to head Everton in front from a cross by Ademola Lookman in the 87th minute and two minutes later fellow substitute Tosun capped a counter-attack with a powerful finish past Palace goalkeeper Wayne Hennessey.
"It's devastating to lose in the last minutes of the game," said Palace defender Aaron Wan-Bissaka. "We played well and created chances in the second half."
"The plan was to defend first, not concede and take our chances, but we didn't take enough. We didn't play badly, we will learn from this game.
"We stick together and keep fighting," he added.
Coleman, returning from injury after missing Everton's four previous league games, had the first chance at Goodison Park on Sunday but shot over the crossbar.
Palace then started to get into the game, with James Tomkins heading wide from an Andros Townsend cross.
Pickford turned away a Milivojevic free-kick before Cheikhou Kouyate went even closer to a goal when his header hit the bar.
Following a goalless first half, the match sparked into life in the second period.
First came Palace's penalty heartbreak, which was soon followed by Hennessey also saving with his feet to deny Everton's Theo Walcott.
But rather than settle for a point, Silva brought on Tosun for Bernard and then introduced Calvert-Lewin and Lookman.