Everton suffered a second successive defeat at Turf Moor despite Cenk Tosun’s first goal for the Club.
The Turkey international confidently headed the Blues in front on 20 minutes after the fit-again Seamus Coleman flicked on a Theo Walcott cross.
However, second-half strikes from Ashley Barnes and substitute Chris Wood gave Burnley all three points in freezing Lancashire.
Stand-in skipper Ashley Williams was shown a straight red card late on following a penalty-box tussle with goalscorer Barnes.
Manager Sam Allardyce made three changes to the Everton side that started at Watford the previous weekend.
Coleman returned from injury at right-back at the expense of Jonjoe Kenny, Tosun replaced Oumar Niasse up front, and Dominic Calvert-Lewin came in for Wayne Rooney, who started on the bench.
The Blues were forced to weather an early storm as Burnley fired in a succession of dangerous crosses during a lively opening.
However, once the game began to settle down, the Blues grew in confidence and they created a golden chance to take the lead in the 13th minute.
Calvert-Lewin played in Gylfi Sigurdsson, who promptly glided past a couple of defenders inside the area before cutting the ball back towards the penalty spot. Walcott and Tom Davies were both lining up to score but they got in each other’s way and the former blasted his shot wastefully over the bar.
Burnley responded with two quick chances of their own, Jack Cork firing over from 12 yards out following a cross from former Toffees winger Aaron Lennon, before Barnes just failed to get a touch on a raking cross that flashed across the face of Jordan Pickford’s goal.
It was proving to be an end to end encounter, with both sides determined to play on the front foot, but it was Everton who ultimately drew first blood in the 20th minute.
Sigurdsson, who was at the heart of most of the Blues’ best work, started the move that led to the deadlock being broken when he fed the ball out to Walcott on the right.
The England winger then whipped over a cross which was nudged on by Coleman to Tosun, who rose majestically to plant a bullet header past keeper Nick Pope from 10 yards out. It was a stunning goal and the first time Everton had taken the lead away from Goodison Park since the 1-0 win at Newcastle United in December.
Walcott’s electric pace was troubling the Burnley defence and the former Arsenal man had two chances to extend the Blues’ lead but on both occasions his finishes lacked the conviction to beat Pope.
At the opposite end, the Clarets continued to look threatening and only a magnificent save from Pickford denied Barnes what looked a certain goal – the Blues keeper brilliantly diving to his right to tip aside a point-blank header.
Pickford was seizing the chance to impress England manager Gareth Southgate, who was looking on intently from his seat in the Bob Lord stand, and he once again showed his mettle when he bravely dived at the feet of Ben Mee in another goalmouth scramble.
Burnley came out fighting after the break and Pickford was soon in the thick of the action again, diving full-length to repel a snapshot from Lennon.
Then, from the resulting corner, the Everton stopper reacted brilliantly to palm away Mee’s close-range header.
Eventually, the Clarets’ pressure told as they grabbed an equaliser in the 56th minute.
Matt Lowton threaded a cleverly-weighted pass through Everton’s defence for Barnes, who raced in behind Michael Keane and kept his cool to beat Pickford at his near post.
Determined to wrestle the momentum back in the Blues’ favour, Allardyce responded by introducing Rooney, and then Niasse, from the bench in place of Tom Davies and Tosun.
Those changes appeared to have the desired effect as Everton - beaten 2-1 on their last visit to Turf Moor in October 2016 - went in search of a winner.
First, Niasse sent an acrobatic volley over the bar. Then Sigurdsson brilliantly wrong-footed a defender in the area but dragged his shot wide of the far post.
However, just as Allardyce’s men were threatening to get back on top, they were undone by a sucker punch, Wood rising highest to meet a corner and plant a header past Pickford from inside the six-yard box.