Home form is becoming something of an issue on Merseyside: neighbours Liverpool have lost six in a row, Everton have won just one of the last seven in which they have been beaten by bottom-six sides Newcastle, Fulham and now the Clarets.
With fourth-placed Chelsea drawing at Leeds earlier in the day Carlo Ancelotti's side had the chance to move within two points with a match in hand but they blew it as they were not only out-fought but out-played for large parts as the visitors' deserved victory moved them seven clear of the bottom three.
Chris Wood and a superb strike from the excellent Dwight McNeil put them 2-0 up inside 25 minutes and it could easily have been more and even Dominic Calvert-Lewin's 14th Premier League goal of the season failed to spark a reaction.
To add more injury to the pain of defeat Everton lost England number one Jordan Pickford to the recurrence of a rib problem.
Burnley, who also ended Liverpool's long unbeaten home run in January, were terrific in the first half as they refused to be destabilised by Richarlison's early shot which Nick Pope pushed behind and Everton's domination of possession without going anywhere and completely against the run of play the visitors took a 13th-minute lead.
Josh Brownhill, excellent throughout, dispossessed Tom Davies deep inside his own half and sent McNeil racing down the left and his deflected cross was clipped home past an unsighted Pickford by Wood.
It was Wood's 40th Premier League goal for the club, just one behind the only other player to reach that tally team-mate Ashley Barnes, and the first time he had scored in back-to-back games this season.
Everton had won just one of their last 24 home league matches after conceding the first goal and they could easily have found themselves 4-0 down in half an hour.
Visibly aggrieved by what they felt was the clear denial of a penalty when McNeil's corner hit the raised elbow of Mason Holgate as he challenged Ben Mee Burnley's intensity increased and they added a brilliant second moments later.
McNeil, in the inside-right position, easily skipped past the tackle of Allan and advanced unchallenged to curl a superb left-footed effort into the top corner.
Johann Gudmundsson then hit the post with a low shot from distance with Pickford aggravating the problem which saw him miss four matches last month with his despairing dive.
Calvert-Lewin got the hosts back on track in the 32nd minute powering home a close-range header from Davies' cross for his first goal in five matches and he should have done better with his flicked attempt from Andre Gomes' free-kick.
Ancelotti had started with Alex Iwobi at the top of a midfield diamond, having said pre-match he wanted the player to tell him his favoured position, but that experiment was soon abandoned as Everton switched to 4-1-3-2 with the Nigeria international moved to the right.
Pickford eventually succumbed to his injury and was replaced by Joao Virginia for his Premier League debut, two and a half years after joining from Arsenal, just before the break.
Like the first half Everton tested Pope early after the interval, the England goalkeeper tipping Gomes' shot around the post.
However, Burnley still looked to have the greater goal threat with a mix-up between Virginia and Godfrey almost gifting Matej Vydra – who later hit a shot too close to the goalkeeper – a chance while Mee's header dropped onto the top of the crossbar.
Virginia denied substitute Jay Rodriguez's angled shot 10 minutes from time but as Everton pushed for the equaliser Burnley dug in and did what they do best, organising, closing space and committing to every tackle.