The Lancashire club are still waiting for an elusive first Premier League victory since they were promoted from the Championship last season after their latest defeat at the hands of Roberto Martinez's side.
Samuel Eto'o put Everton ahead just four minutes in, but Danny Ings scored his maiden Premier League goal to peg the visitors back.
Burnley were caught out again before half-time when Romelu Lukaku, who was at fault for the equaliser, finished a flowing move to put the Merseyside club back in front.
And Eto'o sealed the points five minutes from time with a stunning long-range strike to heap more misery on the home side, who have scored only five Premier League goals in nine games this term.
Burnley will slip to the foot of the table if QPR secure at least a point against Aston Villa on Monday and Dyche knows his team must be more ruthless if they are to avoid an immediate return to the second tier.
"It was a very challenging game against a fine side. Decided, more or less, on high-quality players doing what high-quality players do and that's their clinical edge against ours." the Burnley manager said.
"They showed more sign of it over the game. The general performance was good I thought, particularly first half. I thought some of our football was very good.
"I thought the will and demand to play, but also take chances, was there. But that clinical side is definitely where it's at for us at the moment.
"It's important we keep learning, keep building and keep moving forward. That's the challenge we've got.
"The margins are fine - we've got to find a way of altering it slightly to find a way to win."
Burnley face another daunting fixture next weekend, when they travel to Arsenal.
