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.
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.