The East Lancashire club, promoted last season despite an inexpensive budget, were undone by Kevin Mirallas' 29th-minute strike at Goodison Park.
Burnley have 26 points from 33 matches and next weekend face a resurgent Leicester City that have won their last three matches to climb to third from bottom.
“The loss leaves us fighting,” Dyche told the BBC. “Written off before the start, still being written off now.
“We have had plenty of knocks this season and bounced back. We will look to do that again against Leicester.”
Burnley were reduced to 10 men in the first half when Ashley Barnes was sent off for two bookable offences but Dyche was unhappy Everton forward Mirallas was not dismissed for a rash challenge on George Boyd.
“I am amazed he stayed on the pitch,” Dyche said. “You should be sent off for those.
“You try and do things in the correct manner (not surrounding referees and going down) and it doesn't work for you.
“People want to see people fighting in a fair and proper manner. We try to do that.”
Everton's Ross Barkley missed a first-half penalty and manager Roberto Martinez defended the England midfielder's decision to take the kick ahead of regular taker Leighton Baines.
“Ross showed incredible responsibility to take it,” Martinez said. “I love seeing players take control of the big moments and Leighton was happy to give him that responsibility.
“If Romelu Lukaku had been on the pitch, he would have taken it. Otherwise I am happy to have three or four players who can take penalties and let it depend on how they feel at that moment.”
Everton, unbeaten in their last five league matches, are 12th with 41 points from 33 matches.
Original source: Burnley down but not out yet