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.
9. Ashley Westwood 5.5
Contained Doucoure and Allan in the middle of the park before the break and dropped deep to switch play and provide a link between defence and attack. However, his control of the midfield diminished when Benitez introduced Gomes.
10. Ashley Barnes 5
Occupied Everton's backline alongside Wood, particularly in the first half, which allowed team-mates to advance up the pitch, plugging the space between midfield and attack. However, created very little and lost Keane for Everton's equaliser.
11. Chris Wood 5.5
Failure to convert sublime McNeil cross early on was inexcusable. Started to make amends when dominating Godfrey on the outside of Everton's three, with a superb touch setting up Brownhill's chance on one occasion. In no man's land for Everton's equaliser.