Ben Mee put the visitors deservedly in front at Goodison Park before a three-goal blitz in six minutes just after the hour continued Rafael Benitez’s strong start as Everton boss.
Michael Keane’s header against his old club equalised before two of Benitez’s signings Andros Townsend and Demarai Gray turned the game around.
The Toffees have taken 10 points from their opening four league games under the Spaniard to move alongside local rivals Liverpool, Chelsea and top-of-the-table Manchester United.
Everton’s comeback was all the more impressive as they had to cope without Dominic Calvert-Lewin.
Benitez confirmed before kick-off the England international is set to be absent until October with a broken toe and a thigh injury.
The home side lacked the focal point Calvert-Lewin normally offers as Burnley had the better of the first-half.
Chris Wood and Mee both missed good chances, while Jordan Pickford denied Ashley Westwood.
Mee did not make the same mistake eight minutes after the break when Johann Berg Gudmundsson crossed and the Clarets captain stooped to nod home from close range.
But Burnley then capitulated as Keane was afforded far too much space to head home Townsend’s cross.
Many Everton fans were sceptical about Benitez’s appointment due to his past as a Champions League winning manager for Liverpool.
However, his smart recruitment and tactical acumen is already winning over doubters.
A change of system as Andre Gomes replaced Ben Godfrey straight after the equaliser helped the hosts overpower Burnley in the final half hour.
Townsend previously played under Benitez at Newcastle and displayed his ability to do something spectacular with a long-range effort that looped over Nick Pope into the top corner.
Gray then scored his third goal in as many games by racing onto Abdoulaye Doucoure’s brilliant pass to slot past Pope.
Gomes should have made it four when he fired wide from another Doucoure through ball, but three goals were enough for another three points.