Chris Shuker gave Vale a ninth-minute lead when his free-kick from outside the left edge of the penalty area evaded everyone as the ball floated inside Lee Grant's left-hand post.
But Burnley were level within a minute when the unmarked Chris McCann headed in Ross Wallace's right-wing corner from six yards.
There were further chances at both ends in a lively first-half before a poor back-pass by Vale defender Joe Davis handed the visitors a 29th minute lead.
Dean Marney rounded Vale's stranded goalkeeper Chris Neal to tee-up Charlie Austin who rolled the ball into an empty net from the edge of the penalty area.
And Marney took advantage of more flimsy defending in the 42nd minute when he waltzed into the danger area to lash a left-foot shot high inside Neal's left-hand post from 14 yards.
To their credit, Vale refused to roll over in the second-half as Shuker and Louis Dodds (twice) came close to reducing the arrears with Burnley content to shut up shop.