The Clarets seemed to be cruising when Ben Mee marked his 200th Premier League appearance for the club with the opening goal.
The skipper got on the end of Johann Berg Gudmundsson's pinpoint cross to head the ball beyond England international Jordan Pickford from close range.
However, for the third time in four games this term, Sean Dyche's side let their advantage slip, adding to similar sequences against Brighton and Leeds United.
Ex-Burnley defender Michael Keane equalised against his former club on the hour after heading home Andros Townsend's cross at the near post.
The Toffees went ahead five minutes later when Townsend scored the goal of the game when picking out the top corner from 25 yards out.
With the visitors in disarray, Rafa Benitez's men continued to capitalise on the holes in their opponents' formation.
Abdoulaye Doucoure claimed his second assist of the evening when feeding the ball into Demarai Gray, who tucked the ball past Nick Pope.
Here are the scores on the doors.
1. Nick Pope 6
Made a good save with a strong hand to deny Doucoure in the first half and utilised his height to keep out Townsend's strike after Everton had taken the lead. Stood very little chance of preventing any of the home side's finishes.
2. Matt Lowton 6
Contained the forward runs of Digne and Gray for much of the game, forcing the latter to switch infield on occasions. Everton's productivity was at a premium down that side of the pitch. Got forward well himself in support of Gudmundsson.
3. Charlie Taylor 5
Always carries a threat going forward and forced a number of corners for the Clarets. Made a good recovery challenge in the first half, but all three goals originated from his flank. Failed to put pressure on the ball when Townsend scored the goal of the game.
4. James Tarkowski 6
His thunderous challenge on Richarlison has divided opinion, though it was met with rapturous applause by the away fans. Solid at the back, aside from a tumultuous seven-minute spell, and unfortunate to see his second half header disallowed.