The Toffees went into the match having won their last seven Premier League home games, whereas Burnley were without an away win all season, but any ideas that the result was a foregone conclusion looked to be unfounded when Burnley more than held their own throughout.
However, after Sam Vokes had provided the perfect response to Phil Jagielka's opener, a quickfire double from Everton courtesy of an unfortunate Ben Mee own goal and Romelu Lukaku strike sealed the points for the home side to lift them above Manchester United and Arsenal in the table.
Lukaku went into the match having scored more goals than Burnley's entire team since the turn of the year, but the Belgian squandered a glorious chance to add to his league-high tally as early as the fourth minute when he fired an effort straight at Tom Heaton after being picked out by Holgate.
It was an early chance for the Toffees following a rousing minute's applause on the 28th anniversary of the Hillsborough tragedy, but it was Burnley who made the brighter start to the contest against an unusually sluggish and sloppy Everton.
Heaton was still the busier of the two goalkeepers in the opening 15 minutes, though, with the Burnley shot-stopper also forced to deny Idrissa Gueye while Joel Robles was only faced with a simple stop from Joey Barton's long-range effort.
The Clarets were almost rewarded for their positive start in the 17th minute when PFA Young Player of the Year nominee Michael Keane peeled off at the back post to plant a header towards goal that beat Robles but was cleared off the line by Ross Barkley.
Burnley won another corner moments later and this time played it short, working space for Robbie Brady to curl an effort over the crossbar from the edge of the area.
Sean Dyche's side had been written off by many ahead of the trip to Goodison Park having failed to win a single away game all season, but they were arguably the better team in the opening 45 minutes and could have taken the lead when Vokes raced onto a long ball forward before seeing his effort denied by Robles.
The Burnley striker had another good chance moments later when he latched on to Jagielka's poor back-pass, but he chose to take the shot on first time and could only fire a tame effort straight at the Everton keeper.
Those two chances did seem to spark Everton into life, though, and Heaton needed to be sharp to turn a Mason Holgate effort around the post before Lukaku and Gueye both failed to hit the target from promising positions.
Heaton was as much a spectator as everyone else when the Toffees almost broke the deadlock with just two minutes remaining until half time, but the Burnley keeper was relieved to see Kevin Mirallas's deflected free kick fly inches past the top corner.
Everton broke the deadlock just one minute after that, with Ashley Williams flicking the ball on for Jagielka, whose header just crossed the line before Heaton could get to it as the Toffees skipper made it three goals in his last three games - the best scoring run of his career.
Rather than being deflated by such a poor start to the second half, Burnley responded very well to falling behind and were level again inside three minutes as Vokes blasted home from the penalty spot.
Referee Mark Clattenburg had been given little choice but to award the penalty after a moment of madness from Robles, who brought Vokes down inside the area with a rash challenge despite the Burnley striker moving away from goal at the time.
It was a perfect reaction for a side who have amassed a league-low three points from losing positions this season, but Everton were soon back on the front foot and Mirallas squandered a big chance to put his side back in front shortly before the hour mark when he applied a tame finish to Barkley's through-ball.
The Belgian had another good sight of goal moments later when the ball dropped back to him after he had scuffed his initial effort, but his second bite at the cherry crashed against the woodwork from a tight angle.
Everton's second goal did finally arrive with 20 minutes remaining, though, and it was a difficult one for Burnley to take as Barkley's strike took deflections off both Keane and Mee on the way through before looping over the despairing dive of Heaton.
Barkley peeled away to celebrate a positive end to a difficult week, but it was later confirmed that it would go down as a Mee own goal.
There was no doubt about who the third Everton goal belonged to, though, as Lukaku capped off the scoring just four minutes later by using his strength to turn Keane before smashing an emphatic finish past Heaton from a difficult angle.
The goal saw the Belgian become the first Everton player to score in nine successive home games since Dixie Dean in 1934, hitting 14 goals in that time to continue his march towards the Premier League's golden boot.
That was enough to kill off Burnley's hope of an unlikely point as their woes on the road continued, making it now 19 games without an away win across all competitions this season and just four points from a possible 51 on the road in the Premier League.
The defeat also sees the Clarets drop down into 14th place, although they remain eight points clear of danger with just five games of the season now remaining.
Everton, on the other hand, have now won eight consecutive home games for the first time in their Premier League history, scoring 29 goals in that time - more than any other side in Europe's top five leagues since the turn of the year.
The Toffees now sit fifth in the table, although they could drop back below both Manchester United and Arsenal when they play the first of their three games in hand later this weekend.