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.
5. Ben Mee 6.5
Marked his 200th Premier League game for the Clarets with the opening goal, heading past Pickford from close range. Then, from the restart, made a fine block to deny Allan. Pulled here, there and everywhere when Burnley lost their way after the hour.
6. Johann Berg Gudmundsson 5.5
Worked hard alongside Lowton to shut the door down Burnley's right hand side, which had kept Digne and Gray quiet. Didn't offer a great deal going forward, but his pinpoint cross led to Burnley's opener at Goodison Park.
7. Dwight McNeil 6
Burnley's chief source of creativity in the first half, alongside Taylor. Often showed Allan a clean set of heels when drifting infield. Provided a peach of a cross for Wood early in the game, which was crying out for a finish, but the striker failed to oblige.
8. Josh Brownhill 5
Should have at least tested Pickford when picked out by Barnes in the first half. Received a booking for dragging down Townsend after the break. Poor pass led to Everton's second goal and struggled to compete following the home side's formation switch.