Pte Sean Benton, 20, from Hastings, was found with five bullet wounds to his chest in June 1995 while on guard at a perimeter fence at Deepcut.
An original inquest into his death recorded a verdict of suicide.
His family applied for a new hearing after using the Human Rights Act to access evidence held by police.
His twin, Tony Benton, and sister, Tracy Lewis, wept and said: "We are just happy and relieved. It's been too long."
A spokesperson for the Ministry of Defence (MoD) said: "We care deeply about our young recruits and our thoughts remain with the family and friends of Private Sean Benton."
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The malevolent culture at Deepcut barracks
Justice Collins said fresh evidence had "come to light which casts some doubt upon the correctness" of the original decision.
He added that under article two of the Human Rights Act there was material that questioned the care Pte Benton received from the Army at the time.
Mrs Lewis said: "Our family had just 20 years with Sean. It has taken us another 21 to secure the thorough, independent inquiry we should have seen immediately after his death."
"For two decades, our family has been tormented by questions about what Sean went through at Deepcut," she added.
"If his death had been properly investigated in 1995, we would have been spared years of uncertainty and pain.
"It should be a source of huge shame to the Ministry of Defence and Surrey Police that our mother had to fight for so long - far longer than she should have had to - to force the authorities to answer basic questions."
The family said the original inquest took less than two hours and heard from six people. A criminal investigation seven years later found no evidence of third party involvement.
Pte Benton's medical records were not obtained and no evidence was gathered or presented about his experiences at the base.
The family, who are represented by human rights group Liberty, believe he was subjected to severe bullying.
In a statement, the MoD said: "The Armed Forces will not tolerate abuse, bullying, or discrimination. All allegations are taken very seriously and will be thoroughly investigated, either by the civil or military police and, where appropriate, action will be taken."
"It would be inappropriate to comment further while legal proceedings are ongoing -  we will assist the coroner as necessary," it added.
Pte Benton was the first of four young soldiers to die at the barracks between 1995 and 2002.
Pte Cheryl James, 18, from Llangollen, North Wales, shot herself in November 1995, according to a second inquest into her death which concluded in June. However, the coroner criticised a lack of training, structure and inappropriate sexual relationships in the camp.
Pte James' father, Des, said he had applied to the MoD for a full public inquiry into what went on at the barracks.
Pte Geoff Gray, 17, from Hackney, east London, was found dead with two gunshot wounds to his head on 17 September 2001.
Pte James Collinson, 17, from Perth, was found dead with a single gunshot wound through his chin on 23 March 2002.
A date for the fresh inquest into Pte Benton's death is yet to be set.
A fresh inquest can take place into the death of a young soldier at an army barracks in Surrey 21 years ago, a high court judge has ruled.