PHOTOS: Burnley 4, Port Vale 2
Clarke Carlisle gave the Clarets the perfect start heading in Chris Eagles’ cross after just five minutes.
Port Vale equalised when after a scramble in the box, the ball landed at the feet of Robert Taylor who poked home.
Tyrone Mears restored the lead for the Clarets when his 20-yard free-kick beat Stuart Tomlinson in the Vale goal.
After the half-time break, Burnley extended their lead, when a quick flowing move allowed Jack Cork to play in Chris Eagles who shot first time powerfully past the Vale keeper.
A Graham Alexander penalty settled the tie after Tyrone Mears was upended in the box, Alexander smashed the ball low and hard, sending Tomlinson the wrong way.
Robert Taylor managed a consolation for Port Vale, pouncing after Grant punched the ball out to his feet.
Burnley will play hosts to Burton Albion at Turf Moor in the fourth round.