Cenk Tosun scored his first goal for the Toffees to send them into half time with a slender lead, but Burnley deservedly turned the game around in the second half courtesy of goals from Ashley Barnes and Chris Wood.
A miserable afternoon was compounded for the out-of-form visitors when Ashley Williams capped off a poor personal performance with a red card in the closing stages as Everton were left with only one win from their last 23 Premier League away games.
The result sees Burnley reach the 40-point mark in the Premier League table courtesy of their first win since mid-December, whereas Everton remain in ninth and have now lost their last six away games across all competitions.
Everton boss Sam Allardyce dropped Wayne Rooney to the bench and handed rare starts to Tosun and Dominic Calvert-Lewin, but it was Burnley who made the brighter start to the match and drew the first save from Jordan Pickford within five minutes when Johann Berg Gudmundsson's shot deflected off Seamus Coleman on the way through.
The first big chance of the match fell the way of the visitors, though, when some dazzling footwork from Gylfi Sigurdsson saw him reach the byline and pull the ball back for Theo Walcott, who could only lift his finish well over the crossbar when he should have scored.
Burnley were soon back on the front foot, though, and Jack Cork snatched at one half-chance inside the area before Pickford got an important touch to Gudmundsson's low cross into the area.
It was Everton who broke the deadlock against the run of play, though, as Tosun opened his Toffees account at the fifth attempt with a firm downward header into the bottom corner after Coleman's flick had taken Walcott's cross into his path.
The £27m January arrival almost made it two in as many minutes moments later as Everton suddenly seemed to have been jolted into life, but this time Nick Pope was able to keep out Tosun's low effort from the edge of the box.
The chances continued to arrive at both ends in a first half which provided more goalmouth action than many will have been expecting, with a defensive error from the returning James Tarkowski allowing Walcott a half-chance which was thwarted by Pope.
Moments later Burnley were up the other end and it took a stunning reaction save from Pickford to deny a Barnes header from Matthew Lowton's dangerous delivery.
Barnes threatened twice more shortly after the half-hour mark either side of a chance for Walcott, who produced a tame finish after getting on the end of a searching cross-field pass from Tosun.
The Clarets created another clear chance five minutes before the interval too, but Aaron Lennon could not steer his effort on target after losing his marker to get on the end of Gudmundsson's free kick, while Ben Mee was slow to react to the loose ball inside the area.
Burnley boss Sean Dyche brought on record signing Wood at half time, and the hosts came out of the blocks flying for the second half with Lennon forcing Pickford into another smart save within two minutes of the restart.
The resulting corner saw Pickford called into action again when he reacted well to deny Mee's header, before Wood nodded a follow-up effort over the crossbar.
Burnley did get a deserved equaliser shortly before the hour mark, though, as Lowton's defence-splitting pass down the channel released Barnes through on goal, and this time the striker made no mistake with an emphatic finish from close range after Pickford had opted to stay on his line.
Allardyce responded by introducing the experienced head of Rooney, but Burnley continued to swarm all over Everton as they went in search of their first ever Premier League win under Dyche after conceding the first goal.
Tosun did threaten a second when he got his head to Calvert-Lewin's cross from the right, but the hosts were quickly back on the attack and Barnes almost scored a carbon copy of the equalising goal when he was again found by Lowton down the right channel, only for Pickford to this time come out and thwart the striker.
Everton did begin to grow into the second half with around 20 minutes remaining as Walcott's cross forced a goal-saving clearance from Lowton, before Oumar Niasse sent an acrobatic effort well over moments later.
Sigurdsson wasted the Toffees' best chance of the second half, though, sending his left-footed finish wide of the far post after showing good composure to create space for his shot inside the area.
It proved to be a costly miss as Burnley completed their turnaround just two minutes later, and it was no surprise to see it come via the aerial route as Wood planted a firm header home from close range.
Everton's centre-back pairing of Williams and former Burnley man Michael Keane had struggled to deal with Burnley's attack all game, and they were again at fault for the goal as Wood lost Williams and towered above Keane to meet Gudmundsson's corner.
Things would go from bad to worse for the hapless Williams when he was shown a straight red card four minutes from time after referee Chris Kavanagh ruled that he had caught Barnes with a swinging elbow.
That effectively ended Everton's hopes of salvaging something from the game, and the away fans greeted the final whistle with boos as the result piled more pressure on the shoulders of Allardyce, with Everton losing five consecutive away league games for the first time since January 2015.
Burnley, meanwhile, end what was the longest ongoing winless run in the Premier League to celebrate Dyche's 250th match in charge of the club, while it is also the first time in 54 previous attempts that they have won a Premier League game under Dyche after conceding the first goal.
Burnley have now completed the league double over Everton for the first time since 1959-60, with today's victory effectively confirming their place in the Premier League next season.
BURNLEY (4-4-1-1): Pope; Lowton, Tarkowski, Mee, Ward; Lennon, Cork, Hendrick (Wood, 46'), Westwood, Gudmundsson; Barnes
EVERTON (4-2-3-1): Pickford; Coleman, Keane, Williams, Martina; Gueye, Davies (Rooney, 59'), Walcott, Sigurdsson (Bolasie, 83'), Calvert-Lewin; Tosun (Niasse, 68')