Vale held their own in the hard-fought opening exchanges and home winger Enoch Andoh threatened with a 16th minute shot.
But that was a rare moment of promise during a first half which Vale edged despite failing to test visiting goalkeeper Matt Gilks.
Burnley were virtually non-existent as an attacking force in the first period and it took them almost an hour to pose a threat.
George Boyd’s cross from the right led to a fierce strike from Michael Kightly which was blocked by Vale substitute Ryan Inniss.
And Kightly finally forced a save from Jak Alnwick when his 66th minute shot from 20 yards almost caught out the Vale keeper.
The tie finally began to open up with neither side keen on extra-time and it was Vale who made the breakthrough in the 82nd minute.
Vale substitute Sam Kelly, who replaced Andoh just two minutes earlier, whipped in a cross from the left which Gilks could only palm away.
The ball dropped into the path of Moore and his left-foot shot from 12 yards crossed the line despite Burnley’s efforts to clear.
Burnley rallied late on and Boyd almost equalised when his stoppage time shot clipped the crossbar.