It said the married woman was allowed to stay after telling a judge that her affair left her at risk of being stoned to death in her home country.
The woman, who has been given anonymity by a court, is married to a member of the Saudi royal family, the paper said.
The Home Office declined to comment on the case.
The Independent says the woman is one of a small number of Saudi Arabian citizens whose asylum claims are not acknowledged publicly by either country's government.
The woman reportedly began a relationship with a British man - who is not a Muslim - during a visit to London and later became pregnant.
After giving birth secretly in the UK, she took a case to the Immigration and Asylum tribunal, it is claimed.
Under Saudi Arabia's Sharia law system, adultery is punishable by public flogging or execution.
In 2008, the country's courts ordered the execution of 102 people, according to human rights group Amnesty International.
The latest case appears to echo that of another member of the Saudi royal family, 19-year-old Princess Mishaal bint Fahd, who was executed in 1977 after admitting adultery.
When a controversial film about the case, Death of a Princess, was shown in the UK, Saudi authorities responded by expelling the British ambassador in Riyadh.