Pte Dean Hutchinson's parents, Paul and Elaine, and brother, Liam, issued a statement.
Mrs Hutchinson said on behalf of her family: "Dean was a soldier and we will also be immensely proud of the fact he served his country but we feel that he should never have lost his life in the way he did out there in Afghanistan.
"We are pleased that the coroner's conclusion reflects the Army's incompetence in not carrying out the correct procedures to ensure Dean's safety."
Fighting back tears, Mrs Hutchinson added: "God bless both our boys."
A coroner has criticised the Ministry of Defence for a series of failures over the deaths of two soldiers, including one from County Durham, in a fire which engulfed their tent as they slept at Camp Bastion in Afghanistan.
Wiltshire and Swindon Coroner David Ridley said he would be making a "preventing further deaths" report over the case of Privates Rob Wood, 28, and Dean Hutchinson, 23, of Spennymoor.
Mr Ridley recorded a narrative conclusion and listed eight areas where there was either a "systemic failure" or "failure" in the circumstances that led to the men's deaths.
The two soldiers, who served with the Royal Logistic Corps, were killed when fire swept through a logistical centre at Camp Bastion in Helmand on February 14 2011.
There were delays in alerting the military fire brigade because soldiers at the scene did not know the emergency 222 number.
Fire investigators have concluded that the blaze started in the vicinity of the electrical appliances and quickly spread, igniting combustible materials stored nearby.
The inquest heard that both senior commanders and fire safety officers did not know the soldiers were sleeping on duty during night shifts.
Had they known, the fire risk assessment for the tent would have had to have reflected it, with separate sleeping areas and an unobstructed rear exit.
The "unwritten rule" for the troop was that the duty non-commissioned officer should have remained awake while the other soldiers slept.