The hosts fielded new signing Bobby Petta and the former Celtic star lined up on the left wing for his home debut. But it was the visitors who had the first shot at goal when midfielder Paul Bolland fired narrowly wide in the opening minute of the game.
However the home side took control and, in the fourth minute, Petta's left wing cross struck the top of the cross bar before bouncing to safety.
In the 19th minute a free-kick from Craig Hignett was chested down by Matt Clarke and a quick turn from Alun Armstrong brought out a shot that was well saved by County keeper Saul Deeney.
The visitors remained on the back foot and it was no surprise when Darlington took the lead in the 21st minute through defender David McGurk. Petta's corner was knocked on by Armstrong and McGurk headed into the roof of the net at the far post.
County fought back and the lively Glynn Hurst was only denied, in the 37th minute, by a last gasp tackle from Clarke.
Darlington lead at the break and, with the wind in their favour, should have controlled the second period. But in the visitors did not allow them to play and when the going got tough some of the Darlington players were found wanting.
In the 49th minute County went close to an equaliser when a cross from Stefan Oakes found Hurst at the far post, but his poor shot was easily claimed by Sam Russell and then in the 51st min Russell kept out a shot from David Pipe.
Pipe was not to be denied and grabbed the equaliser in the 59th minute when the Darlington defence failed to deal with a long ball forward from Shane McFaul.
Pipe's first effort was blocked by Russell but he ran in to fire the loose ball past the Quakers' keeper.
The home side were all at sea and the crowd were getting at them which resulted in a double substitution when Petta and Hignett were replaced.
The decision by Quakers boss David Hodgson had the opposite effect because within seconds of the move the visitors took the lead with the goal that proved to be the winner. Kelvin Wilson found space inside the area and he rifled the ball home from 12 yards out.