It had looked like Stanley midfielder Peter Murphy had secured a vital win for his relegation-threatened side with an 88th-minute header but Minshull rescued a point.
It had all started so well for Newport as they took the lead from a disputed free-kick after 11 minutes.
Stanley captain Luke Joyce appeared to win the ball but referee Dean Whitestone pointed to a foul just outside the area and Robbie Willmott fired the resulting free-kick low and hard into the bottom corner.
After being under pressure early on, Stanley came to life and equalised when midfielder Murphy received the ball from Danny Webber to the right of the area and fired into the far corner after 23 minutes for his sixth goal of the campaign.
Stanley then took charge and, after Piero Mingoia's goalbound header was tipped over the bar by County goalkeeper Lenny Pidgeley, duly took the lead after 44 minutes.
A deep ball to the far post by Murphy found the head of Kal Naismith, who jumped with Pidgeley and got the final touch to give Stanley the interval lead.
Harsh words must have been said at half-time and Newport piled on the pressure after the break.
They got their rewards when midfielder Ryan Burge got the ball just inside his own half, chested it down and ran forward before unleashing a fierce shot which gave home 'keeper Marcus Bettinelli no chance.
A late Lawrie Wilson corner found the head of Murphy at the far post for his second of the game but there was more drama to come when Minshull stuck out his foot eight yards out to convert Willmott's cross.