Craig Dawson ended Stockport's brave resistance with a late winner to send West Ham into the fourth round.
Non-league Stockport thwarted the Hammers in appalling conditions, but David Moyes' side got the goal they needed with Dawson's 83rd-minute header.
With County tiring, substitute Jarrod Bowen slung in a dangerous cross and Dawson was first to it, guiding his header into the far corner to spare West Ham extra-time and potential penalty shoot-out.
It was cruel on Jim Gannon's spirited side, who bridged the gulf in class with discipline, industry and application, only lacking conviction in the final third when in sight of the West Ham goal.
Stockport, who famously beat West Ham in a League Cup replay at Edgeley Park in 1996, en route to the semi-finals, have endured a turbulent time since dropping out of the league a decade ago.
That was followed by relegation to the National League North, the sixth tier, in 2013, but under new ownership the Hatters are back in the National League, pushing for promotion to League Two.
They are currently fourth in the National League and have set themselves a target of reaching the Championship, a level they were last at 19 years ago, within the next seven years.
County were bidding to become only the third non-league side to topple a Premier League club, joining Lincoln, who won at Burnley in 2017, and Luton, who saw off Norwich in 2013.
Just as it was 25 years ago, when County humbled the Hammers, it was wet and wild in Stockport, the inhospitable weather lending itself to a potential giant-killing and a tough night for the visitors.
Against that backdrop, David Moyes picked a strong side, including first-team regulars Angelo Ogbonna, Declan Rice and Michail Antonio, his selection showing respect to the hosts.
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As expected, West Ham dominated possession from the start, the first chance falling to Said Benrahma, whose shot from Declan Rice's cross glanced a post.
Andriy Yarmelenko got himself into a shooting position, but his effort went wide, as the gulf in class between the sides – separated by 86 places in the football pyramid – began to show.
After 15 minutes, a spell of fireworks, set off right outside the ground in support of a 15-year-old Stockport fan, currently in intensive care after being struck by a police car on Boxing Day, halted the game for a minute.
County's only real chance in the first-half was a snatched effort pulled wide after half-an-hour by Jordan Williams, as the home side settled into a policy of containment.
At the other end, Stockport skipper Liam Hogan executed a perfectly-timed tackle on Antonio, as the West Ham striker looked to go through on goal.
Then it was the turn of Mark Kitching to produce a vital last-ditch challenge, Antonio again the player thwarted.
The relentless rain turned the pitch into something of a quagmire, which played into Stockport's hands, their opponents not used to playing on such a poor surface.
Stockport began to grow in confidence, a Connor Jennings shot blocked by Craig Dawson, the home applying pressure of their own and creating some openings.
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