It was the Championship side who threatened first with Sam Vokes going close twice as he headed against the crossbar from Kieran Trippier's cross before forcing Michael Ingham into a smart reflex save.
The pressure finally told on the Minstermen's defence after 12 minutes as David Jones scored his first goal for the Clarets as he connected with Ross Wallace's corner to find the net.
York finally registered their first chance on goal after 20 minutes with Sander Puri shooting wide from 25-yards before Ashley Chambers and Ryan Bowman both went close.
Burnley doubled their advantage in the 61st minute with Junior Stanislas emphatically converting Trippier's pass beyond Ingham into the bottom corner.
The result was put beyond any doubt in the 77th minute as Danny Ings tapped home after Dean Marney's shot was saved before Scott Arfield broke away and slotted home with nine minutes left on the clock.