Danny Rowe's penalty gave Stockport the three points as Darlington fell into the Blue Square Bet Premier relegation places for the first time this season.
Stockport, who had not won an away match since 18 October, nearly took the lead but Sean McConville's chipped effort bounced off the crossbar.
Darlington's Rob Ramshaw was shown a second yellow card when he fouled Jon Nolan and Rowe converted the penalty.
Rowe could have added a second but was denied by goalkeeper Jordan Pickford.
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Darlington boss Craig Liddle told BBC Tees:
"It was certainly there for the taking, but we didn't have the belief, to be fair.
"We improved when we went down to 10 men, because the pressure was off.
"People wanted to shoulder more responsibility then because we had nothing to lose."