Kathleen Thompson, 47, was shot dead in the back yard of her Rathlin Drive home in November 1971.
Last week, Ms Judge Sandra Crawford ruled that the fatal shots were fired by a person known as Soldier D as the British Army withdrew from the Creggan Southway area following an arrest operation in the area.
The coroner said Ms Thompson went to the back of her house and banged a dustbin lid or other object on the floor to warn people of the army’s presence when she was shot.
She said Soldier D claimed he opened fire after being shot at and therefore acted to protect himself and his colleagues.
The coroner said: “I cannot be satisfied that Soldier D honestly believed he was under fire.”
Over the weekend, people rallied in Guildhall Square, where they stood alongside the Thompson family to acknowledge their victory after decades of campaigning.
Ms Thompson’s daughter Minty told the crowd that after more than 50 years, her mother had finally been found completely innocent.
Ms Thompson, who was just 12 when her mother was killed, said: “On the night she was murdered the British State did not even consider her to be, worth investigating. When they gave us an examination, it was an open judgement. The mother of six was worth £84.07. That was the value they placed on her.
“But the state has forgotten that there is a family behind every murdered person. A family that will fight for truth, recognition and justice. Kathleen Thompson was our mother and we fought and we fought hard but without the support of the people of Derry and across the North we might never have won. But we did it.”
Meanwhile, the Mayor of Derry and Strabane has paid tribute to the “strength, courage and tenacity” of the Thompson family.
Sandra Duffy said: “Her strength and determination have inspired many other families. My thoughts are absolutely with them at this point.”
Her views were shared by her party colleague on the council, Paul Fleming, who praised the family for always fighting with “valor and dignity”.
He added, “It gives hope to others in a similar situation and we know many families who are in this situation and it will give hope to each of them.”
The chairman of the SDLP group on the council, Brian Tierney, also welcomed the Thompson’s campaign for truth and justice.
He said: “I know it’s been a long, hard battle for them and when you look at what they’ve suffered over the last of these odd 50 years, it’s quite a significant day, not just for the Thompson family and for victims across the north but for the people of Derry to stand up and have the strength to speak up about wrongdoing whenever it happens on our streets.”
Alliance Council Member Rachael Ferguson sent her party’s support to the family, noting, “I know that Kathleen Thompson’s family has struggled really, really hard and I wanted to extend our help in every way as she goes forward.”
Maeve O’Neill, Councilwoman for People Before Profit, praised the Thompson family’s “relentless” campaign, saying: “It’s just a shame it took a court over 50 years to say what we all knew and what the Thompson family for 50 years says .
“It’s good to see that day has come.”