County were just five minutes from their first defeat of the season until White levelled after a goalmouth scramble.
Second-placed Darlington appeared to have done enough to knock County off top spot when substitute Guylain Ndumbu-Nsungu headed home Neil Wainwright's cross in the 80th minute.
Darlington boss David Hodgson was disappointed that his side could not hold on for a valuable win.
Akpo Sodje was presented with chances in either half to give Darlington what would have been a deserved lead.
Sodje should have given Darlington a 17th minute breakthrough but he could only fire straight at Kevin Pilkington and moments later, Carlos Logan, who appears to be getting better with every game, showed good skill on the edge of the area before pulling his shot just wide.
Jonathan Hutchinson then came to Darlington's rescue in the 24th minute as he cleared off the line from Lewis McMahon's dangerous deep cross.
Kelvin Wilson then provided the Darlington back line with another wake-up call when, unmarked, the defender poked wide at the back post.
At the other end, after good work by Wainwright down the right, Sodje rose well but could only direct his header over the bar from David Duke's cross.
Darlington were quick to find their shooting boots early in the second half as Sodje toe-poked over the bar after a strong run by Anthony Peacock.
Just as Ndumbu-Nsungu appeared to have done enough to give Darlington all three points, White was quickest to react in the home box to stab the ball past Bertrand Bossu.