The British government immediately said it would appeal the High Court judgement, which came after a two-week hearing in July.
The court heard allegations then that Jane Muthoni Mara, Paulo Muoka Nzili and Wambugu Wa Nyingi were subjected to torture and sexual mutilation.
The trio's lawyers said Nzili was castrated, Nyingi severely beaten and Mara subjected to appalling sexual abuse in detention camps during the Mau Mau rebellion.
Nyingi was caught up in beatings that led to the deaths of 11 men at the Hola camp in 1959, now known as the "Hola Massacre".
A fourth claimant, Ndiku Mutwiwa Mutua, has died since legal proceedings began.
High Court judge Richard McCombe ruled on Friday that "a fair trial on this part of the case does remain possible and that the evidence on both sides remains significantly cogent for the Court to complete its task satisfactorily".
The judge said: "I am justified in concluding that the available documentary base is very substantial indeed and capable of giving a very full picture of what was going on in government and military circles in both London and Kenya during the 'emergency'."
Supporters of the Kenyans wept when they heard the judgement, although the claimants themselves were not in court.
In the Kenyan capital Nairobi, around 150 Mau Mau veterans gathered at the country's Human Rights Commission, singing and shouting as the news broke.
Some of the now frail and elderly veterans men and women performed a shuffling celebration dance.
In London, Martyn Day, the lawyer behind the claims, called it a "massive, massive decision" but urged the government to act quickly as his elderly clients "won't last for much longer".
"It's been a long battle," he told AFP outside the London court. "Many times I thought we'd never reach this point."
A supporter outside the court, Nyanbura Wainaina, whose mother is Mau Mau, said she was "elated".
"The most important thing is that the truth is going to come out at last," she told AFP.
"The British made our people look like barbarians but they were fighting for their rights and their dignity," she added.
The British government accepts that British forces tortured detainees but said it was "disappointed" with the judgment.
It warned of "potentially significant and far-reaching legal implications" but pointed out that Britain's relationship with Kenya had moved on since the colonial days, stressing "our people to people ties are and will remain strong and deep."
The case will have access to an archive of 8,000 secret files that were sent back to Britain after Kenya became independent in 1963.
London had initially argued that all liabilities were transferred to the new rulers of Kenya when the east African country was granted independence and that it could not be held liable now.
But in 2011, a High Court judge ruled the claimants did have a valid case.
The government then claimed too much time had elapsed since the crimes, but Judge McCombe concluded on Friday these claims were "more illusory than real."
The Kenyans are demanding an apology and the establishment of a welfare fund to ensure that around 1,000 surviving former detainees can have some dignity in their old age.
At least 10,000 people died during the 1952-1960 Mau Mau uprising, with some sources giving far higher estimates.
Tens of thousands were detained, including US President Barack Obama's grandfather.