Pope, who came to blows with goalkeeper Chris Neal after Cheltenham's second goal, opened his account in the 19th minute before adding further strikes in the 58th and 74th minutes to cap an inspired display.
The forward’s hat-trick saw promotion-chasing Vale close the gap on leaders Gillingham to five points as they consolidated second spot.
Roy O'Donovan grabbed a late winner as Northampton moved into the third automatic promotion place with a 1-0 victory over Torquay.
Rotherham moved to within a point of fifth-placed Cheltenham as a result, beating AFC Wimbledon 1-0 at home courtesy of Alex Revell’s 76th minute goal.
Southend’s new manager Phil Brown almost enjoyed a perfect start to his reign at Bradford with his side netting two early goals before James Hanson struck seven minutes from time to salvage a 2-2 draw for the Bantams.
Oxford United shared a 1-1 draw with Morecambe at home as substitute Jack Redshaw grabbed a stoppage-time leveller, while Barnet and Dagenham and Redbridge played out a goalless draw at Underhill.
Struggling Accrington Stanley claimed a crucial 1-0 victory at Wycombe as Lee Molyneux scored the game’s only goal in the 90th minute, while Adam Mekki’s late goal earned relegation-threatened Aldershot a 1-1 draw at Rochdale.