The Clarets took the lead on 24 minutes from Wes Fletcher before a Lilywhite equaliser on the half hour from captain George Miller.
The Burnley winner came on 79 minutes from sub Danny Ings. Ings had replaced former PNE player Keith Tracy on 66 minutes.
Burnley top the table with 13 points from 6 matches, PNE drop to second with the same points but have played one game more. Morecambe are third also with 13 points from 7 matches. Tranmere are fourth on 9 points after their 4-1 home defeat to the Shrimps yesterday.