Western powers had condemned last month's takeover and suspended economic assistance to Sudan, which has been trying to recover from a deep economic crisis.
The coup ended a military-civilian transitional partnership, triggering weeks of deadly unrest and demonstrations against the military, which on Nov 12 reinstated Prime Minister Abdalla Hamdok and promised to free all political detainees.
The army's reinstatement of Hamdok faces opposition from pro-democracy groups that have demanded full civilian rule since the ousting of autocrat Omar al-Bashir and have been angered by the deaths of dozens of protesters since the Oct 25 coup.
Faki is the latest of several prominent politicians to be freed following the Nov 12 agreement, which placed Hamdok at the head of a yet-to-be-formed cabinet of technocrats for a transitional period.
Faki had been part of the civilian coalition that shared power with the military following the toppling of Bashir in 2019. He sat on the joint military-civilian Sovereign Council that served as the country's head of state, and was leader of a committee tasked with rolling back the Bashir regime.
Another committee leader, Wagdi Salih, remains in detention.
Following the coup, other members of the committee as well as dozens of resistance committee and union activists were arrested. While Hamdok has stated the agreement applies to them as well, it is not yet clear how many have been released.
Faki was among several politicians who engaged in a fierce war of words with the military in the weeks prior to the coup.