Burnley's wait for a win away from home continues but the club moved ever closer to sealing their survival in the top flight.
Defeats for both Swansea City and Hull City, who lost out to Watford and Stoke City respectively, means that the Clarets are now eight points clear of the drop zone with just five games of the season remaining.
Everton were chasing a club record eighth straight win at Goodison Park in the Premier League but it was Burnley who had the better of the first half.
Sam Vokes cut a frustrated figure after a sequence of missed opportunities in front of the home support in the Gwladys Street End as he made Joel Robles' job much easier than it should have been.
Burnley boss Sean Dyche was questionned about his side's lack of creativity in recent weeks but there was certainly no shortage of chances in Merseyside.
Joey Barton's attempt from distance, comfortably dealt with by Everton's Spanish stopper, wasn't the away side's most clear cut but it got the ball rolling.
Ronald Koeman has spoken of his admiration for Michael Keane on numerous occasions this term, with the England international reportedly a target for the Toffees in the summer, but there wouldn't have been any appreciation from the Dutchman had the defender netted.
From Robbie Brady's corner, earned after the record signing's set-piece had been deflected wide, the 24-year-old's towering header had Joel beaten only for Ross Barkley to intervene by clearing the ball off the line.
With the Clarets salivating at the prospect of a first away win of the campaign, sensing the opportunity to cause an upset, they continued to press and probe.
Vokes, Brady and Ashley Barnes all bust a gut to meet Matt Lowton's delightful delivery but neither of the trio were able to make a connection to make Joel work.
The former Atletico Madrid goalkeeper was tested, though, when Vokes slipped goal side of Phil Jagielka from Barton's beautifully weighted volley, making an instinctive save to keep the scores level.
And Vokes should've been more productive when telegraphing Jagielka's under hit back pass, failing to capitalise when tamely side-footing the ball at Joel.
The half was by no means one-sided, although Burnley offered their most complete and balanced performance away from Turf Moor to date. They carried a goal threat while maintaining a characteristic discipline at the back.
Everton, however, had their chances to make a mark before the interval. Romelu Lukaku had the first chance of the fixture in the opening five minutes when collecting Mason Holgate's pass inside the penalty area but the striker stroked the ball straight at Heaton.
After denying Idrissa Gana Gueye's right-footed strike, the Clarets skipper then plunged low to his right hand side to get a fingertip to Holgate's attempt which seemed destined for the bottom corner.
Lukaku sent a header in to the stands with the break looming while a section of the stadium were on their feet as Kevin Mirallas's free kick, which took an awkward touch off George Boyd, dropped agonisingly wide of the upright.
Koeman introduced Enner Valencia for the second half and the substitute almost had an immediate impact when his stinging drive was beaten away by Heaton at his near post.
All Burnley's efforts petered in to insignificance, though, when Everton got their noses in front minutes later.
Leighton Baines's corner was helped on by both Valencia and Ashley Williams and Jagielka, unmarked from close range, threw himself at the ball to guide it over the line, via the upright, despite Heaton's best efforts.
The visitors could have folded at that point. But, as you would expect under Dyche, they didn't. Infact they were level within three minutes.
Joel committed a needless foul, inexplicably upending Vokes while the Welshman had his back to goal in an unthreatening position from Jeff Hendrick's pass.
Vokes picked himself up from the turf to tuck the ball in to the corner from the spot, sending Joel the wrong way in the process.
Everton responded immediately and Mirallas squandered a big chance as the Belgium international failed to finish from Barkley's incisive pass.
But Burnley's resilience unravelled shortly after Mirallas thumped the post from close range.
Barkley, who had endured a nightmare week in the build up to the fixture, punched in a bar before being attacked in Sun columnist Kelvin MacKenzie's disturbingly damaging and distasteful piece, put all that behind him when his shot took a nick off both Keane and Ben Mee and the post on its way in to restore the home side's lead in the 71st minute.
And it became 3-1 in the 74th minute when Lukaku expertly span Keane from Baines's pass before stabbing the ball home at the near post.
The Clarets did continue to create as Brady's free kick dipped narrowly wide of the post while Barkley, in the wrong place at the wrong time earlier in the week, was in the right place at the right time to once again clear off the line, this time denying Stephen Ward's looping header.