Earlier this year senior judges in Belfast dismissed a bid to overturn a landmark ruling that the prohibition is unlawful.
But Attorney General John Larkin QC, appearing for the Minister, on Friday began a process expected to see the challenge come under consideration by the Supreme Court in London.
Mr Larkin asked the Court of Appeal to certify a number of questions for further determination.
He told the court: "We would suggest there are few,or indeed any, cases which present such a range of important issues."
His request was declined despite acknowledgment of the significance being placed on the case.
Lord Chief Justice Sir Declan Morgan, sitting with Lord Justices Girvan and Coghlin, said: "We consider we should leave this to the Supreme Court to make a decision."
Despite refusing to certify any questions, the Department's legal team can now petition the higher court directly to hear its case.
We're disappointed that the Minister has taken this decision.
John O'Doherty, Rainbow Project
The move makes the latest twist in a long-running legal battle.
In October last year the ban based on relationship status was held to discriminate against those in civil partnerships and to breach their human rights.
A High Court judge found excluding some would-be parents solely on their relationship status narrowed the pool of potential adopters and could not be in the best interests of children.
His verdict came in a legal challenge mounted by the Northern Ireland Human Rights Commission.
The body is seeking to force a legislative change to bring adoption laws into line with the rest of the UK. Unmarried couples in England, Scotland and Wales can apply jointly to be considered for adoption, irrespective of sexual orientation.
But anyone unmarried in Northern Ireland is currently only eligible for consideration as an individual. Those in civil partnerships cannot apply either individually or as a couple.
Judicial review proceedings centred on the blanket ban imposed on same-sex and unmarried couples by the Adoption (Northern Ireland) Order 1987.
The Commission was backed by an unidentified lesbian woman who wants to be able to adopt her partner's biological child.
The women, who have been in a relationship for four years, also wish to enter a civil partnership to signify their commitment.
In June the Court of Appeal appeared to pave the way for gay and unmarried couples to adopt when it upheld the judicial review verdict.
It ruled that the Department had put forward no justification to exclude same sex couples as parties eligible to adopt as a couple.
We have a process that actually works very well for adopted children.
Health Minister Edwin Poots
Representatives of the Rainbow Project, Northern Ireland's largest lesbian, gay, bisexual and transgender advocacy organisation expressed dismay that the department is now seeking to go to the Supreme Court.
Director John O'Doherty said: "Both the High Court and the Court of Appeal have noted the practice of banning same-sex and unmarried couples from adopting is discriminatory.
"Enough public money has been spent on this fool's errand. The Minister should focus his time on ensuring the best available homes for children in care in Northern Ireland."
Speaking to UTV, Minister Poots said that his actions were not directed to any particular community.
"In terms of Northern Ireland, whenever the consultation on adoption legislation was carried out, 95% of people were opposed to changing the legislation to include this," he stated.
"More recently Queen's University have a report which started in 2003 and was only reported this week, it identified that 99% of children in adopted homes had stability and the reason for that was the rigorous assessment process that took place.
Mr Poots added: "There is a no human right to adopt a child for anyone, the right lies with child and all of the evidence is that children are being identified for suitable homes that exist in Northern Ireland."