Everton won 2-0 on Wednesday, clinching a home victory over their local rivals for the first time since 2010 thanks to goals from Jarrad Branthwaite and Dominic Calvert-Lewin.
While it also succumbed Liverpool's title hopes, the victory highlighted a massive step forward in the Toffees' hopes of staying up.
Everton are now eight points clear of the relegation zone, with four matches left to be played, and the possibility of points to come back should they successfully appeal the second of their points deductions for financial breaches.
Dyche, though, insisted the job is not done as he looks towards another big game on Saturday, when Brentford visit Goodison Park.
"Very pleased with the reaction," Dyche told Sky Sports.
"This was a different game with the organisation and the belief in the organisation. They're a good side, we know that. The commitment from us had to be there. The energy of the side and also the key moments.
"We spoke about the desire and the energy of the team and the physical side of the game. The lads did it on Sunday and they'll have to do it again. That's the challenge.
"We stayed focused tonight and I asked if they could stay with us and they did. Sunday was massive and now it was a massively important derby game. For people like Seamus Coleman, he wears his heart on his sleeve and he's so happy.
"We should be on 41 points and the story's different. Three clean sheets in three home games. The challenge for the players is to go and do it again."
Calvert-Lewin led the line brilliantly, converting Everton's second with a towering header early in the second half. He has scored three goals in his last four Premier League games for the Toffees, as many as in his previous twenty-eight(28) appearances in the competition.
"I thought [Calvert-Lewin] was outstanding tonight," Dyche said.
"We've been trying to get him back to where he needs to be and I thought he was outstanding. He gets a goal. He tried to get it over the line and I love that. The second one is a great header. No, I thought he was outstanding tonight.
"Jordan Pickford made some big saves in the first half. It's rare you stop these sides having chances so you need moments yourself. It's hard to pick out one or two [players]."Calvert image widget
