A home defeat by Barnet on Saturday in front of just 1,463 at the cavernous Northern Echo arena left Darlo 19 points adrift of safety in League Two.
Darlington chairman Raj Singh said: "It is with regret that we're having to part company with another manager during what has been a forgettable season but we felt we were left with no alternative after yesterday."
The Quakers lost Staunton's penultimate game in charge 2-0 at Sixfields and were also beaten at home by Northampton at the end of January.