An 90th-minute Scott Arfield strike condemned the Blues to a harsh 2-1 loss at Turf Moor in a game they had largely dominated but failed to create sufficient chances to earn victory.
Yannick Bolasie had put Everton in a position to go on and win after his first goal for the club cancelled out Sam Vokes's opener but they were stymied by a combination of Tom Heaton in the hosts' goal, determined Burnley defending and poor approach play despite plenty of second-half pressure.
Instead, it was the hosts who plundered three unlikely points when Johann Berg Gudmundsson's shot cannoned off the bar and Arfield turned the rebound past Maarten Stekelenburg.
Ronald Koeman made two changes to the side that drew at Manchester City last weekend, replacing Tom Cleverley with Ross Barkley and Gerard Deulofeu with Kevin Mirallas and Everton made a strong start, with Mirallas testing Heaton inside two minutes after a defensive mix-up but the 'keeper beat his shot away.
Another good move from the visitors ended with a wayward overhead kick by Bolasie before Lukaku had a sight of goal from 20 yards but Heaton turned his shot aside and Stephen Ward slid the rebound behind before Mirallas, who was in any case offside, could pounce.
And the Burnley ‘keeper had to be in the right place again later in the half as Everton continued to press and Barkley cut inside to deliver an accurate shot that the keeper again parried away to safety.
The Clarets had largely been restricted to a direct game reliant on Vokes' aerial prowess but after the big striker's tame shot had been easily caught by Stekelenburg, their best move of the half yielded the opening goal. Arfield skipped easily past Ashley Williams and advanced on goal, scuffing a deflected effort that looked to be dribbling past the post until Stekelenburg put a glove on it and palmed it into the path of Vokes who stabbed it home from close range.
Koeman needed a response from his players and, retaining the starting XI for the start of the second half, he got it in the form of raised intensity and an early shot from Barkley that Heaton claimed down low at his near post and a quick-fire move shortly afterwards where Mirallas' low centre was just too far in front of his compatriot, Lukaku.
The pressure told, though, in the 58th minute as a neat Burnley move was broken up in midfield by Idrissa Gueye and Gareth Barry was released the ball to Lukaku who surged forward with Bolasie on his shoulder. The winger assumed responsibility as the pair drove forward and then lashed a low shot into the far corner of the goal via Heaton's leg to make it 1-1.
The game was almost turned on its head a few minutes later as Everton burst forward again and Seamus Coleman skidded an inviting ball across the six-yard box that seemed destined for Lukaku until Michael Keane toed it behind.
Mirallas's far-post header was then blocked and Heaton gloved a Bolasie cross away from Lukaku's head before Gerard Deulofeu came on for Mirallas with a quarter of an hour to go.
The Spaniard teed up Gueye for a low shot that was easily gathered and he was involved in a sustained spell of Everton possession that saw Coleman drive the ball into the mixer but it was too strong for Lukaku to control and the chance evaporated.
With 10 minutes left, the Blues came as close as they would to winning it when Bolasie cut inside and hammered a dipping shot that Heaton impressively palmed over his crossbar.
And it was back down the other end 10 minutes later when Bolasie himself was robbed of the ball on the edge of his own box and a similar effort by Gudmonsson came off the woodwork to Arfield who sent it back across goal and into the opposite corner to hand Burnley the win.