Cheltenham saw their play-off charge stall while County, who could have taken a big stride out of relegation trouble had they won, needed a goal deep into injury time to salvage a point.
Struggling County committed plenty of resources to attack right from the start but they were caught out by a quick move on the break by Cheltenham with 12 minutes played.
Grant McCann's ball into the box found Steve Guinan and he held it up before finding left-back Craig Armstrong, who had ventured forward. Armstrong's cross was met by a deft header from Kayode Odejayi that landed just inside the far post to register his 11th goal of the season.
Bramble headed wide from a good position for Stockport but McCann almost made it 2-0 when his 25-yard free-kick got beyond the defensive wall and goalkeeper James Spencer was forced to make a full-length save.
Cheltenham had looked in little trouble defensively until the 34th minute when they gave away a needless penalty when captain John Finnigan fouled David Beharall on the edge of the box.
Left-back Mark Robinson made no mistake with the penalty but was promptly booked for an over zealous celebration afterwards.
County then took the lead when Jason Taylor played a diagonal ball across the area and Higgs inexplicably stayed rooted to his line allowing Keith Briggs to beat him with a shot at the near post.
Cheltenham manager John Ward sent on big striker Damian Spencer and he had an immediate impact in unsettling the Stockport defence.
When JJ Melligan's cross was pushed clear by Spencer, his namesake was on the edge of the box for a shot that was deflected into the path of Brian Wilson. The left sided player guided a shot in off at least one defender to make it 2-2.
Cheltenham sensed a winner and thought they had got it when Odejayi was fouled in the penalty area when chasing down a long kick from Higgs and McCann tucked away the penalty with two minutes left.
Yet deep into injury time, County substitute Liam Dickinson appeared on the edge of the box and steered a shot beyond Higgs to send the travelling fans into raptures.