In one of the most important constitutional cases in generations, the three judges ruled that the Prime Minister does not have power to use the prerogative to trigger Article 50 of the Lisbon Treaty to start the two-year process of negotiating Brexit without the prior authority of Parliament.
Brexit Secretary David Davis said he had a "constructive and amicable" meeting with his counterparts from the devolved administrations, as the Scottish Brexit Minister urged the UK Government to provide "greater clarity and transparency" on plans to leave the EU.
She added: "The Scottish government is clear that triggering Article 50 will directly affect interests and rights in Scotland, and triggering Article 50 will inevitably deprive Scottish people and Scottish businesses of the rights that they now enjoy".
The Government hopes the High Court judgement on how Article 50 will be triggered will be overruled when an appeal is heard by the Supreme Court.
While a majority of United Kingdom voters backed Brexit in June's referendum, nearly two-thirds (62%) of Scots voted to Remain.
She insisted the move was not an attempt to "veto" the process of England and Wales separating from the EU.
After some MPs criticised interim Ukip leader Nigel Farage's call for a mass pro-Brexit march on the Supreme Court, Downing Street said people had the right to demonstrate.
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After hearing legal submissions and arguments, the 11 judges will reserve their decision and expect to announce the result early in the new year.
The High Court ruled last week that Theresa May could not trigger the formal divorce process with the European Union without putting the matter before Parliament. However, he said the involvement of Parliament meant handing over negotiations with the European Union to its 650 MPs plus the House of Lords.
And Northern Ireland's top lawyer has said a separate legal challenge to Brexit should "leapfrog" the usual legal process and go directly to the Supreme Court.
She said: "The Tories are seeking to bypass the Scottish Parliament, with no proper scrutiny here in Scotland or in Westminster, and I am not prepared to simply stand by and watch that happen".
The Supreme Court said on Tuesday it had set aside four days, December 5 to 8, to hear the appeal.
Scottish First Minister Nicola Sturgeon on Tuesday pledged that she would do all she can to keep Scotland in the European Union, adding that she was not attempting to block England and Wales' departure.
Britain's prime minister Theresa May said the fundamentals of the country's economy were very strong and affirmed her commitment to strengthening public finances after a report indicated that Brexit will prolong the country's reliance on borrowing.