The Clarets have now claimed seven points from three games since sacking long-term manager Sean Dyche and move to 31 points from 33 games while Everton, who face Liverpool later on Sunday, have 29 points from 31 games.
Czech international Vydra slotted home the winner after Dwight McNeil's perfect pass found Wout Weghorst and the Dutchman squared for a first-time finish from his strike partner.
Earlier, Burnley keeper Nick Pope had done well to keep out efforts from Jonny Otto and Nelson Semedo, while McNeil had tested Jose Sa with a fierce drive.
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