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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A campaign to tackle child sexual exploitation has been launched by Surrey Crimestoppers.
Wednesday morning's weather for the east of the region
Wednesday morning's weather for the west of the region