TALKING POINTS: Everton 1, Burnley 2
And in doing so, the Clarets have now won at Liverpool and Everton in the same season for the first time, after goals from Chris Wood and a glorious Dwight McNeil strike put them in charge.
Dominic Calvert-Lewin pulled one back before half-time, but Burnley managed the game superbly in the second half and could have added to their lead, as they moved seven points clear of trouble.
Here are the talking points:
Sean Dyche is a hard man to please. Although delighted with the performance at Goodison Park, when I suggested it was close to his idea of 'footballing nirvana' - the mixed football so effectively produced by Sir Alex Ferguson's Manchester United sides - he replied that was "maybe over-egging it - we've had some good performances this season, and we weren't too bad the other week at Liverpool, but I know what you mean."
And when asked about Sky Sports' Jamie Carragher's quip, that it was like watching Brazil, with Burnley in their vivid yellow shirts, he smiled: "I think we've got a long way to go to be looking like Brazil, but I was really pleased."
While Dyche's standards are high, this was arguably Burnley's best performance of the season - a display that showed everything that is good about Dyche's side.
They pressed high and forced turnovers that led to big chances, they played through the thirds and created openings, they were a threat when they went direct, they caused problems from set pieces, and on the other side of the coin, they made interceptions and blocks, Nick Pope made saves you'd expect him to make, and, all in all, the Clarets looked a formidable, cohesive unit.
Dyche did concede: "We talk about effective football, and I thought it was, particularly first half.
"The defensive shape was good, against tricky opposition.
"And every time we pressed them properly and nicked the ball, I thought we looked a threat. I thought virtually every time we nicked it and went forward, you felt we could score.
"Today was a healthy balance, defending right, work ethic of the team, the ugly side of the game when we needed to, working diligently from the front, and also the effective football I talk about - we played short, long, on the counter, all the things I think are important for an outfit to win games in the Premier League."
Burnley created a memorable slice of history with victory at Everton - the three points mean the Clarets have won at both Anfield and Goodison Park in the same season for the first time!
Remarkably, they are the third Premier League side to achieve that feat this season, after Manchester City and Fulham.
But while Goodison hasn't always been a happy hunting ground - the Clarets had lost five of their previous six visits in the Premier League - they have taken a liking to the Toffees since Wade Ellliott's winner against David Moyes' Everton at Turf Moor in 2009.
This win means Burnley have won six of their 14 Premier League matches against Everton - the most wins against any Premier League opposition, other than Bournemouth and Crystal Palace (also six).
ON THE RADAR
Dyche said of Sky man of the match Dwight McNeil: "I think he continues to develop under the radar, but he certainly won't be under the radar tonight after that goal!"
The 21-year-old earned lavish praise from the Match of the Day team, and across the national newspapers, for a performance which is a new benchmark for the wing wizard.
McNeil is expected to be in the England squad for the European U21 Championships group stages over the international break, but if he builds on this performance, he will be knocking on the door of the senior squad.
It has been noted elsewhere that he needs to up his output, in terms of goals and assists, but at Goodison, he delivered a performance of real maturity and responsibility.
With the ball, he was deadly, putting over the cross which led to Chris Wood's opener, before exquisitely curling in a second, after sending Brazilian midfielder Allan almost back to Naples with a deft turn.
And without it, he helped set the tone, along with the superb central midfield pair of Ashley Westwood and Josh Brownhill, continually winning the ball high up the pitch, while also diligently tracking back.
A lot is expected of McNeil, even at 21, and he appears to set very high standards of himself as well.
But this is the level he must seek to attain on a consistent level - it certainly isn't beyond him.
Burnley had only won once in nine league matches ahead of Saturday evening, but had only lost one in seven. How you looked at things depends on whether your glass is half full or half empty.
But while all the talk has been about Fulham's great form, and how the Cottagers look like hauling themselves out of relegation trouble, Scott Parker's side actually have two points fewer than Burnley since the turn of the year.
The Clarets are 10th in the form table for the calendar year - two behind Spurs in sixth, as they yet again show strong signs of kicking on over the second half of the season.
Before the game, after signing a new contract, right back Phil Bardsley said: “The lads are playing some really good stuff and looking solid as well, so hopefully we can get a couple more wins and get over the line and finish as high as possible.”
Seven points, or a three-result swing - clear of trouble with nine games remaining, victory at Goodison was a perfect way to sign off ahead of a three-week break.
And you wonder whether one more win might just be enough to effectively secure safety now - although, as Bardsley suggests, you can only imagine Dyche's side going hard until the end of a testing season.
Chris Wood made it two goals in two games, with a fine finish, adjusting his body and shaping a shot into the far corner after McNeil's cross was half-cleared to him.
And while Matej Vydra couldn't find the net, it wasn't for the lack of trying.
Wood and Vydra have looked an effective pairing of late, both intelligently setting pressing traps and hunting down the ball, while both being willing runners into the channels.
It has taken time, but Vydra now looks to have completely been assimilated into the Dyche style of play, and his extra pace has allowed Burnley to stretch defences to a greater extent.
It bodes well for his chances of being selected for Euro 2020, where he could line up against England.
And of Wood, you wouldn't bet against him hitting double figures for a fourth-successive season, after making it six for the campaign, with nine games to go.