While there has been an upturn in performances of late, a shambolic opening quarter of the game saw the Clarets trailing 3-0 to an Everton side who barely got out of first gear.
There was a response from Burnley, pulling one back before the break through league debutant Ben Gibson, but they couldn't build up a second half head of steam and Everton added two further goals.
Boss Sean Dyche kept the same 3-5-2 shape used at Arsenal, but made three changes, bringing in Gibson, Matt Lowton and Sam Vokes for Kevin Long, Phil Bardsley and Chris Wood.
But there were only seconds on the clock when the visitors stunned Turf Moor, as a corner was needlessly gifted by Charlie Taylor, and when the flag kick fell to Bernard on the left, his cross was gobbled up by the Colombian centre back Yerry Mina.
Mina showed his prowess from set pieces at the World Cup, including a late equaliser against England, but he didn't have to bust a gut to score on this occasion.
And it was 2-0 20 minutes later as Lowton carelessly clattered into Dominic Calvert-Lewin, and as everyone eyed up Gylfi Sigurdsson taking another expert dead ball, Lucas Digne beat him to it and whipped the ball inside the near post beyond Joe Hart - an exact replica of his late equaliser against Watford.
That looked game over, even then, and it certainly was on 22 minutes as James Tarkowski was too casual on halfway and was robbed by Bernard, who advanced before Tarkowski got back to concede a corner.
From the corner, Ben Mee was harshly adjudged to have handled the ball - Mee certainly did, but wasn't in control, having had his arm in the air to jump, before colliding with Mina.
Michael Oliver pointed to the spot, and Sigurdsson sent Hart the wrong way.
As Burnley continued to struggle to find their way, Calvert-Lewin stabbed wide after appearing to foul Mee.
The Clarets finally came to life when Jordan Pickford over-egged a flick header from Ashley Barnes from an Ashley Westwood free kick, turning the ball behind for a corner.
And from Taylor's dead ball, Tarkowski thought his header had gone in, before Gibson made no mistake, rifling in the loose ball to make it 3-1.
The goal roused Burnley, who ended the half with a good spell of pressure, but
were unable to reduce the arrears further.
At half-time, Everton's three goals meant the Clarets had conceded as many as they had in the whole of last season - 39.
They had to keep the Toffees out after the break and try and score the next goal, and they had a big chance when Tarkowski scooped over in front of goal from Jeff Hendrick's centre.
Barnes then pulled a cross shot across goal, too far ahead of Vokes, while Theo Walcott's low ball in somehow evaded Calvert-Lewin at the other end.
Johann Berg Gudmundsson and Dwight McNeil were introduced for Hendrick and Gibson as Burnley went 4-4-2, but the Clarets struggled to threaten.
Former Claret Keane headed over a Digne corner as Everton looked for a fourth, and they got it when Bernard squared to Digne, whose 35-yard scuffed effort somehow beat Hart to his left.
And it was 5-1 at the death - the seventh Burnley have conceded in the 90th minute or beyond - as substitute Richarlison poked in Sigurdsson's pass.