Everton scored three times in the opening 22 minutes at Turf Moor to end a winless five-match run and push Burnley deeper into trouble.
Any lingering negativity from the Toffees' 6-2 hammering by Tottenham on Sunday was swept aside inside two minutes as Yerry Mina headed in his first goal in English football and Marco Silva's side went on to triumph 5-1.
Lucas Digne curled in a superb free-kick 11 minutes later and Ben Mee's handball allowed Gylfi Sigurdsson to net the third from the spot.
Ben Gibson pulled one back before half-time on his first Premier League start for Burnley following double hernia surgery but any hopes of a second-half comeback were ended by Digne's second in the 71st minute, with Richarlison adding a fifth in injury time.
Everton matched Burnley's three centre-backs and it was the man who came into the team who was celebrating with only a minute and 36 seconds on the clock.
The Toffees took a corner short and, although Mee got a decent head on Digne's cross, Burnley's defence failed to reorganise and Bernard's ball in from the left picked out the unmarked head of Mina six yards out.
Bernard was preferred in the starting line-up to Richarlison, and Burnley struggled to deal with the little Brazilian, who had a shot deflected over the bar by Gibson a couple of minutes later.
The Clarets' afternoon was about to get worse as, after Matt Lowton was penalised for a foul on Dominic Calvert-Lewin 25 yards out, Digne stepped up to send the ball flying inside Joe Hart's near post.
The Frenchman is fast establishing himself as a free-kick specialist, with this effort very similar to the one he scored against Watford earlier this month.
Sean Dyche could point to the fact Everton were raiding Barcelona for defenders in the summer while his signing came from Middlesbrough, but the Clarets were making basic errors and getting punished.
James Tarkowski almost paid for giving the ball away to Bernard on the halfway line but sprinted back to tackle the winger just as he shaped to shoot.
But Everton did score their third with only 22 minutes on the clock. Burnley's fans protested Michael Oliver's decision to penalise Mee for handball as he attempted to clear Bernard's corner under pressure from Mina and Kurt Zouma, but the defender's reaction, as he sat with head in hands, told its own story.
Sigurdsson coolly rolled in the spot-kick as the massed ranks of Everton fans in the away end celebrated exuberantly.
Burnley had barely threatened but they made their first real moment of danger count eight minutes before the break.
Jordan Pickford did well to claw away Ashley Barnes' flicked header and then keep out Tarkowski's effort from the corner, with a bit of help from his post. But the rebound fell for Gibson and he hooked the ball just inside the other post.
Dyche stormed across the pitch towards the tunnel the moment the half-time whistle blew, and he would have been much happier with his team's start to the second half.
Tarkowski should have pulled a second back in the 49th minute as he met Jeff Hendrick's excellent cross six yards out, but he shot over with only Pickford to beat.
At the other end, Calvert-Lewin was inches away from tucking away Theo Walcott's fizzing cross, while Charlie Taylor produced a fine block to deny Sigurdsson after another Bernard run left Burnley's defence caught in the headlights.
Hart then blocked Walcott's effort after more hesitation in the Burnley defence, and it was no surprise when Everton scored their fourth moments later, Digne finding the far corner from distance for the second time in the game.
Burnley tried to mount another comeback but, after recent improvements, this performance was a real step back, and Richarlison added the final nail with a fifth in injury time.
