Mitchell, who is serving a 20-year sentence for killing Jodi Jones, is thought to be the first serving prisoner to release a video online in a bid to win his freedom.
The 24-year-old was convicted of murdering his 14-year-old girlfriend in Dalkeith, Midlothian, in 2003 after one of the most lengthy trials in Scottish history.
He has always maintained his innocence, and is shown in the video – posted on the Luke Mitchell Is Innocent website – answering questions and denying he was responsible for Jones' death.
Watch the video below.
The test – which was sanctioned by the prison authorities – was conducted in Shotts Prison in April 2012 by Terry Mullins, secretary of the British Polygraph Association, who has carried out a string of lie detector examinations in prisons south of the Border. He says Mitchell passed the test.
However, John Lamont MSP, the Scottish Conservative Chief Whip, called the video "insensitive" and a "crass stunt".
He said: "Clearly, there was no thought of how the victim's family would feel when this decision was made. It is insensitive in the extreme and we should not be playing out our justice system on YouTube or any online forum, but through the courts.
"Scotland has a robust appeal system and there should be no place for crass stunts like this."
Mitchell's case is being examined by the Scottish Criminal Cases Review Commission (SCCRC), which investigates potential miscarriages of justice and can recommend a fresh appeal. Polygraph tests are inadmissible in Scottish courts but the report on Mitchell's test was included in his submission to the SCCRC.
Criminologist Dr Sandra Lean, who has spearheaded the campaign to free Mitchell along with his mother, said the lie test provided important evidence.
"The polygraph result in Luke's case reinforces what the other evidence has been telling us all along – there is not a scrap of evidence that Luke Mitchell murdered Jodi Jones," she said.
Mitchell's mother, Corinne, 53, who visited him last Thursday, said: "He says he has never had anything to hide so he didn't hesitate when he had the chance to take it. In fact, he had asked for a polygraph since day one.
"Luke told me he believes the test is a crucial indication of his innocence to go along with other issues, such as the lack of forensic evidence linking him to the crime."
Mrs Mitchell has also passed a lie test, backing her claim she was with the convicted killer at their home at the time of the murder.
Jodi Jones disappeared after saying she was going to meet Mitchell on June 30, 2003. Her body, which had horrific stab wounds, was found behind a wall beside a path near her home in Easthouses.
She had suffered a slashed throat and post-mortem cuts to her eyelids, right cheek, left breast, abdomen and right forearm.
The lie-detector test was the latest move by Mitchell's supporters to have the case re-examined after two previous appeals by his legal team failed.
After the tests were carried out last year, Mr Mullins concluded that Mitchell had been telling the truth.
He said: "I'm certain of the test result. It's absolute. I can't believe Luke Mitchell was convicted on the evidence that was available."
Sources at the Scottish Prison Service said they agreed to give the video to the Luke Mitchell Is Innocent campaign, but did not authorise it being made public.
The source added: "The campaign asked for the video and we agreed to release it. We did not expect to see it surface in this way."
An official SPS spokesman said they would not comment on individual prisoners.