As part of the review, the SPS will consider how safeguards can be put in place to give consideration to the feelings of victims and their families, Alex Salmond told MSPs.
The video of Mitchell, who was jailed in 2005 for the murder of his 14-year-old girlfriend Jodi Jones in Dalkeith, Midlothian, was filmed in Shotts Prison last year and posted on the website YouTube at the weekend.
Mitchell has always maintained he is innocent, and his case is currently being considered by the Scottish Criminal Cases Review Commission after his previous appeal failed.
In the video, filmed by the British Polygraph Association, he is seen denying killing the teenager in 2003.
Responding to a question from Labour MSP Lewis Macdonald about the role of the SPS, and the rules relating to the filming and the release of the video, Mr Salmond said: "Everyone will deeply regret the hurt and upset that irresponsible use of this footage will have inflicted on the family and friends of Jodi Jones."
He told Holyrood that HM Chief Inspector of Prisons Hugh Monro had visited Shotts Prison on Tuesday.
Mr Salmond read from a report produced by Brigadier Monro that stated: "All of this appears to have been carried out correctly, and within prison rules.
"My view would be that the SPS have acted correctly and with good faith. At no stage has the SPS at any level authorised or encouraged the film to be uploaded on to YouTube."
Speaking after First Minister's Questions, Mr Macdonald said: "It is right the Chief Inspector of Prisons has looked into this, but he does not appear to have considered any conditions that could have been imposed on the release of the film, and it is for that reason Jodi Jones's family have suffered.
"It is difficult to understand why this film, with or without conditions, has been released into the public domain. The public interest is not served when convicted prisoners are allowed to broadcast their views in this way and action must be taken to ensure it never happens again."