Gillingham took the lead through Danny Kedwell's far-post header, but Chris Robertson equalised for the home side with a close-range header.
Vale mounted pressure after the interval and it was no surprise when half-time substitute Pope headed home in the 63rd minute.
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Friday evening's weather for the East of the region
The latest weather in the west of the region presented by Aidan McGivern.
The latest weather in the East of the region presented by Aidan McGivern.