With league leaders Gillingham not in action, Micky Adams' side had a chance to move to within five points of the top of the npower League Two table.
But they did not have things all their own way at Vale Park, with Pope's hat-trick only just proving enough to secure the win.
Pope, who has now scored 30 goals this season but had failed to find the back of the net in almost 16 hours of football, opened the scoring with a first-half header.
But Cheltenham fought back and were 2-1 ahead less than 10 minutes after the interval as Kaid Mohamed and Paul Benson struck for the visitors.
Pope then struck twice, with a smart volley and a chip over Cheltenham goalkeeper Scott Brown, to grab the points for second-placed Vale.
The result left Cheltenham fifth in the table and Rotherham took advantage of their defeat to move within a point of Mark Yates' side with a 1-0 win at home to Wimbledon.
Rotherham had the better of the play but had to wait until the 76th minute for Alex Revell to score the only goal of the game.
Aldershot moved off the foot of the table after drawing 1-1 at Rochdale, with York taking their place.
George Donnelly gave Rochdale the lead before Adam Mekki levelled from close range to earn Aldershot a share of the spoils and move within two points of safety.
Phil Brown's first match in charge of Southend looked set to be a memorable occasion for the former Preston boss until Bradford hit back to earn a 2-2 draw at Valley Parade.
Gavin Tomlin and Britt Assombalonga scored within a minute to give Brown's side a 2-0 lead after 11 minutes but Capital One Cup finalists Bradford came back into the game with Zavon Hines scoring early in the second half and James Hanson heading the hosts level late on.
A last-minute Lee Molyneux goal gave Accrington a crucial 1-0 win at Wycombe to move Leam Richardson's side five points clear of the drop zone.
Strugglers Barnet secured a point with a goalless draw at home to Dagenham although Edgar Davids' side finished with 10 men after substitute Kyle De Silva was sent off late on.
Oxford and Morecambe also shared a point as an injury-time Jack Redshaw goal earned the visitors a 1-1 draw at the Kassam Stadium following Alfie Potter's opener for United.
In Friday evening's game, Roy O'Donovan grabbed a late winner as Northampton moved into the automatic promotion places with a 1-0 victory over Torquay.