Everton will rue 11 very costly first-half minutes after missing an opportunity to climb one place in the Premier League and move within two points of fourth-placed Chelsea.
Carlo Ancelotti's side have repeatedly demonstrated the capacity to recover from home disappointments this season but after beating Southampton at Goodison Park a fortnight ago will be extremely disappointed to take nothing from this meeting with Burnley.
Everton weren't wanting for effort in their pursuit of an equaliser during a second 45 minutes that quickly settled into a pattern of home pressure, intermittently released by Burnley counters.
Dominic Calvert-Lewin scored Everton's goal after 32 minutes, bulleting a header past Nick Pope, but, by then, Ancelotti's team were chasing the game.
Chris Wood opened the scoring with a neat finish after 13 minutes. Dwight McNeil deposited a glorious strike into the Howard Kendall Gwladys Street net with 24 minutes on the clock.
Everton's night took another unfortunate turn when Jordan Pickford hurt himself diving to try to make a save shortly after McNeill's goal.
The goalkeper gritted his teeth and tried to continue but eventually succumbed two minutes before the break.
Joao Virginia came on for his Premier League debut.
Everton stay sixth, with 10 games remaining, but know they must summon their ability to strike back from disappointment and hit the ground running when returning to Premier League action in three weeks.
All of Everton’s possession in the opening 25 minutes – they had 71 per cent of the ball, primarily funnelling play through Richarlison – counted for nothing.
Twice Burnley advanced in that period – and both times the away side were ruthless.
In the case of McNeil, they were brilliant.
McNeil – whose wicked set-piece delivery had already disturbed Everton – collected a pass infield from Matej Vydra, before exhibiting tremendous close control to evade Allan.
The winger advanced and from 20 yards sent an exquisite strike off his left boot, arcing away from Pickford and into the top-left corner. McNeil couldn’t have made a better job of it if he’d picked up the ball and placed it precisely where he wanted.
Pickford didn’t stand a chance.
The Everton goalkeeper was equally helpless when Wood scored Burnley’s opener.
Josh Brownhill stole possession in midfield, Wood helping the ball on for McNeill to gallop down the left flank.
Michael Keane cleared the cross but only as far as Kiwi Wood, who slapped a shot from the edge of the box into the right corner.
Everton, to this point, had probed, without getting behind Burnley’s backline.
Richarlison was roaming between Burnley’s midfield and defence, collecting possession in promising positions, but the decisive pass or strike was missing.
Pope tumbled to his left to keep out a Richarlison effort and the Burnley goalkeeper smothered at Calvert-Lewin’s feet after the Brazilian overcooked his through ball.
There was an episode on 20 minutes which saw Richarlison skirting the penalty area and wiped out by a combination of McNeill and Ben Mee.
Jon Moss, the referee, played an advantage but Mason Holgate’s cross was repelled.
Ancelotti responded to seeing his side slip two goals behind by dismantling Everton's midfield diamond, in favour of directly matching Burnley’s two wingers.
Alex Iwobi moved to the right side and when he supplied a terrific pass for Calvert-Lewin on the stroke of half-time, the striker showed too much of the ball to Mee to surrender a shooting opportunity.
Calvert-Lewin, though, had struck his 14th Premier League goal of the season by this point.
Tom Davies was furious with himself for conceding possession in the build-up to Wood’s goal.
But Davies is maturing into an excellent and robust footballer.
He surged into space, leaving Wood flat-footed, to receive Iwobi’s pass back from the byline.
Davies’ cross was flat and hard and landed on the forehead of Calvert-Lewin to turn home.
Burnley wanted a penalty when the ball hit Holgate’s arm as the right-back tussled with Mee from a McNeil corner – this at 1-0 and straight after Ben Godfrey deflected behind a Wood strike.
Indeed, if Everton were wobbling at any stage, it was during the 10 minutes either side of Burnley’s second goal.
The visitors were the width of a post from going three in front when Johann Berg Gudmundsson’s scudding effort thudded into the woodwork.
And Everton’s problems were compounded when Pickford’s features creased in pain as he made contact with the ground in his effort to reach the Icelander’s low drive.
Pickford lasted another 16 minutes – during which Godfrey blocked goalbound drives from McNeill and Brownhill – before very reluctantly admitting defeat and giving way to Virginia.
The Portuguese immediately saved at his feet from Brownhill – but, in terms of opportunities, the flow was turning in Everton’s favour.
Richarlison tried for an equaliser straight after Calvert-Lewin’s goal but Pope was equal to the bouncing effort.
Calvert-Lewin headed off target from Andre Gomes’s whipped free-kick – and within two minutes of the restart it was Gomes taking aim to force Pope into a flying stop low to his left.
Everton forward Calvert-Lewin then got his timing all wrong when Lucas Digne – still up after delivering a corner – crossed from the right.
Gomes was taking charge of Everton’s free-kicks, with Gylfi Sigurdsson on the bench, and another dead-ball delivery was skewed wide by Keane, contorting his body for contact.
Burnley were increasingly content to fall into a defensive shape, allowing Everton to have the ball and relying on the occasional surge forwards for respite.
Davies saw a route over the top and lifted a beautiful pass across 50 yards for Calvert-Lewin, who rued his misfortune when a perfect first touch counted for nothing after an inadvertent ricochet off his opposite boot.
That was Davies’ final involvement, Joshua King introduced to play on the left and Gomes shifting infield.
Vydra swung a boot and missed the ball after a mix-up between Godfrey and Virginia – and the same Burnley player was denied by Virginia’s low save minutes later.
Meantime, Godfrey recovered to make a critical interception with Wood poised to meet an Erik Pieters cross – Mee heading the resulting corner from McNeill against the top of the bar.
Ancelotti made his final roll of the dice with 17 minutes remaining, bringing down the curtain on Holgate’s 100th Everton appearance and sending on Seamus Coleman for his first football since last month’s Merseyside derby.
King, Richarlison and Calvert-Lewin hustled and pressured, trying to suffocate Burnley – but, chasing an equaliser, Everton were vulnerable.
Virginia saved strongly at his near post from substitute Jay Rodriguez – paving the way for Everton to launch a late assault on Burnley’s goal.
Iwobi and Coleman had shots blocked – the game now being contested exclusively in the visitors’ half – and Calvert-Lewin was a centimetre from making contact with Gomes’ low ball across the box.
James Tarkowski cleared with Richarlison looming at the beginning of three added minutes.
Virigina joined Everton’s attack for the corner, which Burnley scrambled away despite Richarlison wanting a penalty after tumbling amid a frantic penalty-box scene.
Everton host Manchester City in an FA Cup quarter-final tie in seven days, Crystal Palace arriving for a league fixture a fortnight later.
The focus for Ancelotti and his players, then, will be on relocating the Goodison consistency that was a hallmark of the opening months of the manager's reign, and has suffered without supporters in the ground.