Manus Deery was 15 when he was killed in the Bogside area of Derry.
A barrister for the MoD and PSNI accepted the teenager had not been carrying a weapon and that the shooting breached military guidelines.
The Deerys led a long campaign to uncover the truth.
Helen Deery, Manus' sister, said: "My whole family is delighted and we are all really emotional.
"Manus would have been 60 this Thursday. It's bitter sweet but I'm relishing the fact the truth has been told.
"It's been difficult recently being in court. We also had another brother die in the middle of the inquest. It's a shame he never lived to see today.
"I have the main answers and that's that he was innocent. When I heard the news, I was very emotional. It's a great feeling that his innocence has been recognised.
"My brother was called a terrorist for almost 45 years. We knew he wasn't. He was a beautiful wee boy.
"Every year after he died I dreaded Christmas Eve because I have memories of me and Manus opening other peoples' presents. His innocence will come back to me again this Christmas Eve.
"The soldier that killed Manus is dead. We are going to speak to our solicitor about prosecutions and where we go from here.
"Compensation was never on the cards for my family."
A barrister told the court: "He was behaving lawfully. He was innocent of any wrongdoing at the time of his death.
"We accept that there was a breach of the Yellow Card and that the decision (to shoot) was premature and cannot be justified," he added.
"The MoD accept that Manus was unarmed and did not pose a threat. That is undisputed."
The killing, which took place at the height of the Troubles, is one of a number of controversial security force cases being re-examined by the coroner.
At a fresh inquest last month, the name of the soldier who killed Manus Deery was officially disclosed for the first time.
Private William Glasgow was never convicted and has since died.
The soldier had maintained that he fired at what appeared to be a gunman about 200 metres away, missed, and the stray bullet killed the boy.
The sister of a Londonderry boy, who was shot dead by the Army in 1972, has called for prosecutions after the Ministry of Defence (MOD) accepted his innocence.