High Court judges backed an earlier ruling that Liam Campbell would be held in inhuman and degrading conditions in the Baltic state.
But although they directed that the 49-year-old should be discharged after nearly four years in custody, he will remain behind bars until at least next week.
A lawyer for the Republic of Lithuania confirmed plans to appeal the latest verdict to the Supreme Court in London.
Campbell, of Upper Faughart, Dundalk, was one of the men held liable for the Omagh bombing.
He is wanted in Lithuania over allegations he was part of an operation to acquire guns, ammunition and explosives from there and ship them into Ireland for dissident republicans.
His lawyers resisted extradition proceedings by arguing that it would breach his human rights.
They brought in a special adviser to the British Home Office and the Council of Europe's Committee for the Prevention of Torture (CPT) as part of their case.
Professor Rod Morgan visited Lukiskes Jail in Vilnius and delivered a critical assessment of prison regimes in Lithuania.
Last month, a judge refused to order the extradition after studying details of severely overcrowded and unsanitary conditions at Lukiskes.
He ruled that Campbell would be at real risk of inhuman and degrading treatment by reason of the jail conditions. Counsel for the Lithuanian authorities challenged that decision by claiming it was based on out of date information.
A panel of senior judges was told Prof Morgan's visit was in May 2010, with no evidence of the current position.
However, Lord Justice Girvan held that problems identified by the CPT have not been resolved and, in some respects, are worsening.
He said: "In a continuing situation demonstrating deterioration rather than improvement and an economic situation showing a lack of resources to counter that trend, the common sense inference to be drawn is that the conditions already condemned as inhuman and degrading by Strasbourg still prevail, at least in parts of the prison to which the returned person may very well be exposed."
The judge, sitting with Lord Chief Justice Morgan and Lord Justice Coghlin, held that extradition would infringe on Campbell's human rights.
Dismissing the appeal, he added: "We must accordingly order the discharge of the prisoner."
Campbell has been in prison since he was arrested after crossing the border into south Armagh in May 2009. A month later he was found liable, along with convicted Real IRA leader Michael McKevitt, for the Omagh bombing.
That ruling followed a landmark civil action brought by relatives of some of the 29 people killed in the August 1998 atrocity.
According to the judge at the time there was cogent evidence that Campbell was a member of the Real IRA's Army Council.
Two other men originally held responsible, Dundalk-based builder and publican Colm Murphy and Seamus Daly, from Culaville, Co Monaghan, are currently facing a civil retrial after the findings against them were overturned on appeal.
On Friday, as Campbell listened to the verdict via video-link from Maghaberry Prison, his barrister Barry Macdonald QC confirmed he would be seeking bail.
But Gerald Simpson QC, for the Lithuanian authorities, argued that the requested person should remain in custody while a further challenge is prepared.
"My instructions are to indicate to your Lordships that we intend to apply for leave to appeal to the Supreme Court," he said.
With the bail hearing adjourned until Tuesday, Campbell will be kept in prison until then.