The Lancashire club slumped to bottom of the Premier League for the first time since November as Kevin Mirallas' strike sealed the spoils for Roberto Martinez's men, with Burnley's troubles confounded by Ashley Barnes' sending off on the stroke of half-time.
But Dyche is adamant that, despite only having claimed one win in their last 12 league fixtures, Burnley still believe they can avoid the drop.
"There's never been a lack of energy with us, it's just a lack of cutting edge, but there's belief, so they'll be ready to go again," he said.
"The loss leaves us fighting, but we were written off before the start of the season and we're still being written off now so that's no change.
"But we've had plenty of knocks this season and bounced back."
Everton set the tempo at Goodison Park and were gifted an opportunity to take the lead when David Jones fouled Aaron Lennon on the edge of the penalty area, with referee Mike Jones pointing to the spot.
However, Ross Barkley's lacklustre strike was well saved by Tom Heaton, although the England youngster's blushes were spared by Mirallas' 27th-minute opener.
Barnes – already on a yellow for a clumsy tackle on James McCarthy – then saw red for a reckless challenge on Seamus Coleman, but Mirallas was lucky to remain on the pitch after a dangerous lunge on George Boyd in the second half.
And Dyche was dismayed by the some of the referee's decisions.
"I don't know how he [Mirallas] stayed on the pitch [for a high tackle]. You should be sent off for them," he added.
"He is not a malicious player, but you can't stay on the football pitch for tackles like that.
"The penalty was, in my view, outside of the box.
"I do think the referee got the sending off right, though, Ashley [Barnes] gives everything for the team, but there has to be an element of control, and he's committed two silly challenges."
