Hamshaw weaved his way into the area six minutes from time, cleverly exchanged passes with Simon Brown and then poked a tame 10-yard finish which somehow squirmed under keeper James Spencer.
Four minutes from time a spectacular save by Spencer denied Stephen Dawson a last-gasp Mansfield winner and then from the following corner Hamshaw's rocket crashed against the post.
The thrilling finish was in contrast to a scrappy first half lit-up by County's first goal of the season courtesy of a close-range Keith Briggs header.
County, without a goal and a point from their first two matches, had changed formation switching to a 3-4-3 line-up after the disappointment of the opening week of the season which left them rooted at the foot of the table.
Manager Jim Gannon dropped strikers Liam Dickinson and Michael Malcolm from the midweek 4-3-3 formation, bringing in Carlisle loanee Glenn Murray and Adam Griffin.
County, wearing their third choice kit after a clash of colours, were almost given the perfect start when Murray blazed a 12-yard volley over the bar from a first minute corner.
Mansfield recovered to pose a threat of their own. Michael Boulding's powerful 17th minute free-kick from the edge of the area fell kindly to Brown inside the six-yard area, but the striker's shot on the turn was acrobatically blocked by Ashley Williams.
And that man Williams then created a 23rd minute breakthrough for the visitors as Murray headed his deep free-kick to the near post for Briggs to nod in from six yards.
Moments earlier the Stags had lost towering central defender Jon Olav Hjelde, and their inability to win the aerial battles then allowed County to dominate the rest of the first half.
Tony Dinning's curling 18-yard free-kick was deflected for a corner as the visitors chased a second goal but they couldn't find a way through.
And in first-half injury-time the Stags almost found a way back but Brown's shot on the turn was blocked on the line by Williams.
Two minutes into the second half a fine stop by Spencer denied Jonathon D'Lareya from close range as the home side poured forward in search of an equaliser.
Then full-back Johnny Mullins glanced Dawson's cross narrowly wide as the Stags laid siege to the Stockport goal and Hamshaw curled a 12-yard shot narrowly wide of the upright.
On a rare breakaway, Adam Griffin lifted his close range shot wide in the 79th minute as County looked to have weathered the home storm.