Abdel Hakim Belhaj and his Moroccan wife Fatima Boudchar say they were kidnapped and taken to Tripoli, where they were tortured.
The claim is against Mr Straw and Sir Mark Allen, MI6's counter-terrorism chief at the time they were taken in 2004, as well as against the Home Office and Foreign Office.
The Government attempted to block the damages claim, but that failed at the Supreme Court on Tuesday.
The couple had offered to settle for £1, an apology and an admission of liability for what they suffered under Gaddafi. Liability was denied.
Lawyers argued they were the victims of a secret rendition in a joint MI6/CIA operation and were entitled to justice for "abuse at the hands of our government".
But the Government had argued that the claims should be barred under state immunity and the "act of state" doctrine - which prohibits the courts of one country sitting in judgment of the acts of the government of another within its own territory.
The High Court agreed with the Government's argument and ruled the claims should be struck out.
But the Court of Appeal reversed that ruling in October 2014, with appeal judges saying there was a "compelling public interest" in allegations of unlawful rendition and "particularly grave violations of international law and human rights" being investigated by English courts.
Now seven Supreme Court justices have upheld that decision and the action can go to trial.
Sapna Malik, representing Mr Belhaj, said: "The Supreme Court today has delivered an emphatic judgment upholding the rule of law, particularly in the face of breaches of rights recognised as fundamental by English statute and common law, in which British defendants are alleged to have been complicit.
"We hope that the defendants in this action now see fit to apologise to our clients and acknowledge the wrongs done, so that they may turn the page on this wretched chapter of their lives and move on."
Human rights organisation Reprieve said Mr Belhaj and his wife, who was pregnant, were kidnapped in Malaysia, abused and forcibly transferred to a Libyan prison in 2004.
It said that in 2012, the UK Government settled a similar claim relating to the apparent kidnap and rendition to Libya of a family that included four children under the age of 12.
In a statement, Mr Straw said: "At no stage so far have the merits of the applicant's case been tested before any court. That can only happen when the trial of the action itself takes place.
"I repeat what I said in the House of Commons in December 2013, that as Foreign Secretary I acted at all times in a manner which was fully consistent with my legal duties, and with national and international law.
"I was never in any way complicit in the unlawful rendition or detention of anyone by other states."