The visitors took the lead after just 14 minutes when Chris Birchall ran at the Shortwood defence before moving the ball on to Jennison Myrie-Williams, who drove his low strike across the keeper and into the bottom corner.
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King was called into action again in the 26th minute when he parried Lee Hughes's stinging volley away after the hosts could only half-clear a long ball forward.
The second goal came on the half-hour mark when Pope got on the end of Hughes's flick on. The striker's effort was saved by King, but he could do nothing about Birchall tapping home the rebound from four yards.
Port Vale came within inches of extending their lead even further in the 72nd minute when Pope rose highest in the box to direct a header goalwards, but his effort came off the underside of the bar.
The visitors did make it 3-0 with just two minutes of normal time remaining as Myrie-Williams netted his second goal of the match from the penalty spot.
They added a fourth two minutes into stoppage time to complete the scoring, as Lines nodded Myrie-Williams's cross past King to end Shortwood's remarkable cup run in emphatic fashion.