Stockport suffered a fourth successive home league defeat as Yan Klukowski won the game for Forest Green.
After the start was delayed because of traffic problems, Stockport dominated the early stages of the game with Michael Paton going close twice.
But Klukowski got the only goal with a calm finish after James Norwood had got to the byline and pulled the ball back.
Norwood nearly doubled the lead with an 18-yard shot but Stockport's Matt Glennon made a smart save to deny him.
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Forest Green Rovers manager Dave Hockaday told BBC Radio Gloucestershire:
"It was a long, great day and a fully deserved victory. I'm very proud of everybody. The fans, my players and my staff were all super. We've come to Stockport County and we've deserved to win.
"We should have won by a lot more, as we had a lot of chances and they didn't test my keeper at all.
"We've defended resolutely today and counter-attacked brilliantly, I thought they were the better team until we scored and that hurt them."
Stockport 01 Glennon 02 Lynch (Whitehead - 61' ) 05 O'Donnell 22 Piergianni 24 Edwards 04 Routledge Booked 08 Nolan Booked 16 McCann 10 Miles (German - 46' ) 19 Chadwick 23 Paton (Fraughan - 69' ) Substitutes 12 Ormson 25 Blackburn 11 Fraughan 06 German 15 Whitehead Forest Green 01 Bittner 02 Hodgkiss 03 Stokes 05 Turley 22 Oshodi Booked 08 Klukowski (Thomson - 64' ) 11 Forbes 19 Rowe 20 Bangura 07 Norwood 10 Griffin Substitutes 12 Imudia 16 Graham 04 Turk 21 Thomson 14 Taylor