The prognosis had looked quite worrying at half-time when the Toffees overturned Nathan Collins’ first goal for the Clarets to take control.
Richarlison netted twice from the spot, sending Nick Pope the wrong way on both occasions, after Ashley Westwood had brought Anthony Gordon down in the 18th minute and then Aaron Lennon tripped Vitalii Mykolenko four minutes before the break.
However, the home side responded and found themselves back on level terms just before the hour when Charlie Taylor did exceptionally well to get away from Jonjoe Kenny to pick out Jay Rodriguez for the equaliser.
And just when it looked as though Sean Dyche’s side would have to settle for a share of the spoils, substitute Matej Vydra laid on the winner for Maxwel Cornet with just five minutes remaining.
Here are the ratings.
1. Nick Pope 6
Sent the wrong way by Richarlison from the penalty spot on two occasions, but made a crucial save to keep out the Brazilian's acrobatic effort with the score locked at 2-2. Photo: Laurence Griffiths
2. Connor Roberts 4
An off night for the Welshman, who couldn't get a handle on Richarlison. The Brazilian had the right back's number throughout, beating him with the ball at his feet and with runs off the ball, and he became a target for the Toffees as the game wore on. Replaced by Lowton with 20 minutes remaining. Photo: James Gill
3. Charlie Taylor 7
Produced an exceptional turn to show Kenny a clean pair of heels before firing the ball across the penalty area for Rodriguez to equalise. Enjoyed a competitive tussle with Gordon, closing the space to limit the 21-year-old's productivity. Photo: Stu Forster
4. Nathan Collins 6.5
Marked his return from suspension with the opening goal, turning home Cornet's corner at the far post after 12 minutes. Unconvincing defensively at times, conceding possession cheaply while adding a few botched clearances, but closed the ball well when Everton moved the ball around the penalty area. Photo: Lewis Storey