County started the game looking to take full advantage of a lack of confidence in a Brentford side that was hoping to bounce back from a shock FA Cup defeat at Telford.
It was almost a dream start for the home side when Tommy Wright was cautioned for fouling Andy Welsh, which allowed Ricky Lambert to line up one of his trademark free-kicks but his effort flew wide off the target.
Jim Goodwin then missed a 29th minute penalty. Bees keeper Paul Smith made the save to his right and County loan signing Simon Lynch, screwed the rebound past the post.
County were looking to secure only their second home league win of the season, but the Brentford back three stood firm.
It all went wrong for Stockport on 66 minutes when Danny Jackman side-footed Wright's cross into his own net to put the visitors ahead.
That spurred the home side to push forward in numbers and in search of the equaliser that their approach play deserved, therefore leaving gaps at the back.
And it was only a last-ditch tackle by John Hardiker in a two on one situation that prevented Brentford from doubling their lead.
Just as it looked as though Brenford would register their first win in six games, Stockport scored with eight minutes left on the clock, substitute Stuart Barlow sending a looping header over Paul Smith.
Stockport manager Sammy McIlroy commented: "It was a below par performance, but at least the point got us out of the bottom four." Bees boss Wally Downes was happy, adding: "It was not a great spectacle, but we had to do something after the defeat at Telford.
"We changed our system and I am pleased with the team's overall effort."