The club, who played in Division Three (North), eventually went out to Liverpool after defeating Billingham, Nottingham Forest, Barnsley and Hull.
Stockport have moved on since that 1949-50 season despite living in the shadow of the two Manchester clubs, United and City. Under rookie manager Andy Kilner they have preserved their status in the First Division, although the next couple of months could be crucial to their survival hopes.
Tomorrow, however, their league campaign will be forgotten as they face Tottenham Hotspur in the cup at White Hart Lane. Kilner's troops have a chance to emulate yesterday's heroes by reaching the quarter-finals but he is aware they face a daunting task.
While Spurs boss George Graham is high-profile, Kilner could probably walk through Stockport's Merseyway shopping precinct without being recognised. The paradox is that Stockport are a club that embraces its community in every sense of the word.
When they sign on, players know that they have to get involved in coaching and encouraging different age groups, be it men, women or children. Should they fail to carry out their duties on a designated day they are fined. There is no comeback as community work is specified in their contracts.
Kilner made such a good job of the Football in the Community Scheme he was promoted to club manager in June 1999 when Gary Megson departed.
While Graham enjoyed a marvellous playing career at club and international level, Kilner's was ruined by serious injury.
One of the few people to have played for England youth at football and cricket, his future seemed assured not matter what sport he took up. He played alongside the likes of Tony Adams and Tim Flowers and made his debut for Burnley on the left wing under manager John Bond at age of 17.
But then fate struck the Swinton-born youngster a cruel blow. He ruptured his medial ligament and lacerated his cartilage in a routine A team game. Kilner tried to pick up the threads in Sweden with Halmstad, but his bad luck continued and he suffered a badly broken leg.
After he recovered he had spells with Vanerborg and Jonsered before returning to England with Stockport. He helped them get out of the Fourth Division, his powerful shooting and all-action displays, making him a fans' favourite. They nicknamed him "Killer" and it's a tag that has stuck.
But was off-loaded by manager Danny Bergara, who decided to dispense with wingers, in 1993. Loan spells at Rochdale and Bury followed then a stint in Norway with Frederikstad. However, his old injuries resurfaced and he turned to community coaching first with Bolton.
Kilner has boosted his squad in the last 20 months with two surprise arrivals at Edgeley Park, Finland internationals Jarkko Wiss and Shefki Kuqui. Just this week Kilner invested another £150,000 to bring Glynn Hurst from Ayr, although he is ineligible to play at Spurs.
Perhaps the key man against Tottenham will be former Derby favourite Kevin Cooper. He is a mirror image of Kilner in his playing days - a raiding winger with a sweet left foot.
"He's fast becoming the most sought-after player outside the Premiership," said the Stockport boss.
While Kilner has strengthened his playing staff, the club has extended its community arm and potential fan base - all the way to China. A delegation led by chairman Brendan Elwood leave on Monday for a 12-day visit.
Stockport have plans in the pipeline to play three or four friendlies in China at the end of the season. The club are also aiming to invite 25 children to spend eight weeks in the town.
For the moment, however, Stockport's thoughts are on the capital, not China, and the possibility of a famous victory in north London.
You have to go back 51 years to the last time unfashionable Stockport County reached the fifth round of the FA Cup. Then King George VI was on the throne and Clement Atlee was Prime Minister.