Everton moved out of the bottom three for the first time this season, having thrown away the lead in their past three games.
They were 2-0 up against both Bournemouth and Aston Villa, before being beaten 3-2 both times, while they were held to a 1-1 draw at promoted Leicester last Saturday.
But this result against Palace – and the return of star defender Jarrad Branthwaite – will provide hope that they can now kick on this term as they host Newcastle next Saturday, followed by a trip to Ipswich.
“It is a step forward,” said Dyche. “The work here is never done, that is for sure. The reality is to go and win again and that is the mentality we have to show. We park this quickly and get on with it next week and build up to the next game.
“I wish we had won all five [previous league games] but the reality it is a tough league. It is good for the players, they have earned the right from previous results because there are a lot of question marks to play in a nervy situation and see the game through properly like we know we can do and find the moments that can win a game.
“I don’t think we have been as far off as people think, especially the last few performances but you have to get things right in both boxes and we did that today.”
Palace, meanwhile, remain winless after six games this term and manager Oliver Glasner said: “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.”