Sean Dyche expressed his relief after Everton earned their first three points of the season through a Dwight McNeil brace at Goodison Park. (More Football News)
The hosts conceded early as Marc Guehi put Crystal Palace a goal to the good after 10 minutes, as Everton were again undone following a set-piece.
But a fine second-half display from McNeil saw two excellent finishes carry Everton over the line.
It was the first time this season Dyche's side had managed to maintain a lead, with the Toffees having dropped points from winning positions in their last three Premier League matches.
Following news of a possible takeover for the Merseyside club this week by The Friedkin Group, who also own Roma, Dyche was relieved to secure his first victory of the season, and just a sixth league win of 2024.
"I'm delighted for the players, they worked very hard to change a scoreline around," the Everton manager told BBC Sport.
"You know there’s been question marks over many things, and of course, there's been a bit of relief there for me today and relief for the players, and that’s a human thing.
"There's a lot of demand here that we put on ourselves and we have to answer to that and I thought we did today. We certainly did enough to win the game and that’s what was important."
Everton are yet to keep a clean sheet in the Premier League this season, though Jarrad Branthwaite's return from injury was a major bright spot.
And the Everton defence did look stronger despite a shaky first-half display. The visitors accrued an expected goals (xG) total of just 0.85 from 17 shots.
"It was a bit of nervousness. When you haven’t got that first win you feel nervous. Then you go 1-0 down, the stadium becomes awkward for the players," Dyche explained.
"I spoke to them in the second half about being someone who makes a difference. It’s every person that counts, not just one."
Everton proved far more clinical thanks to McNeil's long-range effort on 47 minutes. They finished on 0.91 xG from just eight shots.
"I thought the feel of the second half was better, not just the quality. Dwight popping up with two goals for two different reasons and some good energy in our play and the big moments went for us today," said Dyche.
Crystal Palace also had the better share of possession with 59.4%, a source of disappointment for boss Oliver Glasner.
"Over 70-75 minutes we controlled the game, but you must be focused all game. It is a big strength of Everton's in set-plays. They do it well and we defended it most of the time really well," he told BBC Sport.
"We tried everything, we changed the system, brought on all our offensive players. They defended with 10 players around their box so we didn't have a lot of space so it was difficult.
"Today we couldn't score the equaliser. If you concede two goals it's not so easy to get a point or to win."
It leaves the visitors without a win in their last six games in the Premier League.
"We know it when we look at the table. Maybe it is a welcome back to reality. Maybe we were flying too high after a great finish to the season. The Premier League is a fight," Glasner added.
