Goals in either half from Jarrad Branthwaite and Dominic Calvert-Lewin on Wednesday sealed a first victory over Liverpool at Goodison Park since 2010.
The Blues are now on a run of three straight home wins which has now helped open up an eight-point gap on the Premier League’s relegation zone.
And Dyche was quick to credit the Goodison faithful for their support during the 244th showdown which also ended a three-year wait for a derby success.
He said: “I’m delighted with the performance of our players against a top side – they are a top side, we know that.
“I am delighted for Evertonians, for the old school Toffees who support us.
“People do forget, among all the question marks, there are a lot of people when I am out and around they do understand some of the challenges we have had.
“[They said] ‘we know what you are doing, we know you are trying to work hard to change all the storylines’, so thank you to them for listening.
“I thought they were terrific again tonight, I thought they were terrific on Sunday, after a really tough week for me, the team and the fans.
“I only asked them to stay connected, I said ‘just give us your lot, we will try to give it back’, and we have done and I am pleased for the club at large.”
