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.
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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.