Pope, who was voted the best player in League Two by rival managers last weekend, netted a priceless treble to take his season's tally to 30 goals.
Vale recovered from the early loss of striker Lee Hughes to a hamstring injury to dominate most of the first half and take a 19th-minute lead.
Cheltenham were caught napping from Jennison Myrie-Williams' short corner and a terrific cross from Dan Jones led to Pope burying a downward header inside Scott Brown's left-hand post from 10 yards.
Brown then kept Town in the game with some terrific saves before Paul Benson served notice of the visitors' revival when he headed against the crossbar from Billy Jones' long ball.
Town boss Mark Yates made two changes at half-time and one of the substitutes, Byron Harrison, played a major part in the equaliser just 72 seconds after the interval.
Harrison's shot was blocked by Vale 'keeper Chris Neal but only into the path of Kaid Mohamed who fired the ball into an empty net.
And Cheltenham went ahead seven minutes later when Neal kept out Michael Hector's firm header from Mohamed's corner but Benson scrambled in the rebound.
It was now a thrilling contest with a chance at one end or the other every minute and Vale levelled just before the hour mark when Pope rammed in Myrie-Williams' left-wing corner with a fierce low shot from 10 yards.
And it was Pope who had the final say in the 74th minute when he galloped clear from Louis Dodds' pass to fire an expertly placed left-footed shot inches inside Brown's post from 14 yards.