The Quakers were at the bottom of the table before Staunton, who was assisted by another former Sky Blues Kevin Richardson, arrived.
Less than 1500 fans were at the Darlington Arena to see them beaten by Barnet on Saturday and the Darlo board made the decision to end the former Republic Of Ireland's manager's contract.
Craig Liddle and Neil Maddison have been put in caretaker charge of the team which looks to be heading towards the Conference.