Communities Secretary Sajid Javid told BBC One's Question Time the referendum result had been "very, very clear" and politicians should "get on with that".
But three judges ruled that the PM could not trigger the formal Brexit process without Parliament's backing.
The Supreme Court is expected to hear the government's appeal next month.
Prime Minister Theresa May is due to speak to European Commission President Jean-Claude Juncker by phone on Friday to say she intends to stick to her March 2017 deadline for triggering Article 50.
Addressing Thursday's ruling on Question Time, Mr Javid said the vast majority of MPs had backed legislation to bring forward a referendum on the UK's EU membership in 2015 and the public had clearly voted in favour of leaving the EU.
He said there was a "moral issue" at stake: "It was a clear result, clear instructions were issued... by the British people to their politicians saying: We need a decision. Now it's our job as politicians to get on with it."
He said he was not criticising the judges, but the people who brought the case: "This is an attempt to frustrate the will of the British people and it is unacceptable."
But for Labour, Lisa Nandy told the programme: "Britain is leaving the EU and whether or not Parliament has to vote to trigger Article 50, this will happen because, in reality, there are no more than a handful of parliamentarians who would seek to block that decision."
In the landmark ruling, the High Court judges said the government could not trigger the Article 50 process of formally leaving the European Union alone - they must have the approval of Parliament.
BBC political editor Laura Kuenssberg described the ruling as a "massive obstacle" for Mrs May, who says she wants to trigger Article 50 by the end of March.
If the government loses in the Supreme Court, it will have to publish some form of new law for MPs - and the House of Lords - to vote on. MPs could then push to set the terms for negotiating withdrawal and it could face further delays in the Lords.
The prime minister's spokeswoman has played down suggestions that Mrs May may call an early general election if she cannot get Parliamentary support, saying that Mrs May believed "there shouldn't be an election until 2020 and that remains her view".
Brexit Secretary David Davis told the BBC the result of the EU referendum "must be respected".
"Parliament voted by six to one to give the decision to the people, no ifs or buts, and that's why we are appealing this to get on with delivering the best deal for Britain."
But Gina Miller, the lead claimant in bringing the case to the High Court, urged the government to "do the responsible sober thing, which is to do the job we pay them for, to debate all the aspects to do with leaving and then have a vote".
By Eleanor Garnier, BBC political correspondent
It is one of the most important constitutional court cases in generations.  And the result creates a nightmare scenario for the government.
Theresa May had said she wanted to start Brexit talks before the end of March next year but this ruling has thrown the prime minister's timetable up in the air.
Campaigners who brought the case insist it was about "process not politics", but behind the doors of No 10 there will now be serious head-scratching about what the government's next steps should be.
This decision has huge implications, not just on the timing of Brexit but on the terms of Brexit. That's because it's given the initiative to those on the Remain side in the House of Commons who, it's now likely, will argue Article 50 can only be triggered when Parliament is ready and that could mean when they're happy with the terms of any future deal.
Of course, it will be immensely difficult to satisfy and get agreement from all those MPs who voted to remain. Could an early general election be on the cards after all?
She rejected criticism that the case was subverting democracy and said it was "not about politics, this was about process".
"One of the big arguments [in the referendum[ was parliamentary sovereignty... so you can't on the day you get back sovereignty decide you're going to sidestep or throw it away."
The government had argued it could use ancient prerogative powers to give effect "to the will of the people".
But the three judges looking at the case found there was no constitutional convention of the royal prerogative - powers used by ministers - being used in legislation relating to the EU.
They added that triggering Article 50 would fundamentally change UK people's rights - and that the government cannot change or do away with rights under UK law unless Parliament gives it authority to do so.
But UKIP leader Nigel Farage said he feared a "betrayal" of the 51.9% of voters who had backed leaving the EU in June's referendum.
"We are heading for a half-Brexit," he said.
"I worry that a betrayal may be near at hand... I now fear that every attempt will be made to block or delay the triggering of Article 50."
Labour leader Jeremy Corbyn said: "This ruling underlines the need for the government to bring its negotiating terms to Parliament without delay. Labour respects the decision of the British people to leave the European Union. But there must be transparency and accountability to Parliament on the terms of Brexit."
Liberal Democrat leader Tim Farron told the BBC: "What the government was seeking to do was to impose a deal that absolutely nobody voted for, that most people who voted to leave wouldn't be happy with and most people who voted to remain wouldn't be happy with, without any kind of Parliamentary scrutiny.
"So it's a terrible shame it had to go to the courts, but what has happened is that democracy and sovereignty have been restored."
The UK voted by 52% to 48% to leave the European Union in a referendum on 23 June.
The EU's other 27 member states have said negotiations about the terms of the UK's exit - due to last two years - cannot begin until Article 50 has been invoked.
What questions do you have about the Article 50 High Court ruling?
A High Court case that could delay the Brexit process is "a clear attempt to frustrate the will of the British people", says a cabinet minister.