Counsel General Mick Antoniw called the ruling "a victory" in upholding the sovereignty of Parliament.
But the court dismissed Welsh ministers' argument the assembly should be consulted on starting Brexit.
Giving the judgement on Tuesday, President of the Supreme Court Lord Neuberger said: "On the devolution issues, the court unanimously rules that UK ministers are not legally compelled to consult the devolved legislatures before triggering Article 50.
"The devolution statutes were enacted on the assumption that the UK would be a member of the EU, but they do not require it.
"Relations with the EU are a matter for the UK government."
The Welsh Government had argued that if MPs did not vote on Article 50, it would undermine the basis for devolution.
The Welsh Government had also argued that using royal prerogative powers to trigger Article 50 short-circuited the Sewel Convention, which requires Welsh assembly members to vote on legislation which affects the Welsh devolution settlement.
The judgement said that the Sewel Convention was a political constraint on the activity of the UK parliament, but that the policing of its scope and operation was not within the constitutional remit of the courts.
In December, the UK government appealed against a High Court ruling that Parliament must approve the Article 50 process of leaving the EU.
However, a majority of the judges dismissed the argument that the prime minister could trigger Brexit using crown prerogative - powers used by ministers that do not require the permission of MPs.
Mr Antoniw had earlier argued the Article 50 process also needed a vote in Cardiff Bay.
He told BBC Wales the ruling was "certainly a victory in terms of upholding the sovereignty of Parliament", and that process now "opens the opportunity to engage through the Sewel convention".
"We've never argued for a veto, and the court made that point, but what it does do is stress the importance of the Sewel convention in terms of engagement."
Plaid Cymru spokesman for external affairs Steffan Lewis said the party would still seek to table a legislative consent motion in the assembly on the triggering of Article 50.
"It is a simple matter of democracy that the devolved legislatures should have a role in commencing the process of leaving the EU," he said.
UKIP MEP and independent AM Nathan Gill said the Brexit white paper published by the Welsh Government on Monday "will make it into the Guinness Book of Records for having the shortest shelf life ever, 24 hours".
"Because it's now been made obsolete," he claimed.
But Mr Gill - a member of the committee advising First Minister Carwyn Jones on Brexit - said the Supreme Court judgement in favour of parliament was "no surprise".
A UKIP assembly group spokesman said it welcomed the judgement on Article 50 and Parliament, and that any attempt to block Brexit would trigger an immediate general election: "We say bring it on."
"It would be absurd for Wales, Scotland, Northern Ireland to have a veto over triggering Article 50," he said.
The Welsh Conservatives' Europe spokesman Mark Isherwood, said: "The Welsh Government's tortuous arguments were an expensive sideshow.
"It would have been better if every effort had been focused on delivering a Welsh Government paper on Brexit before the UK government announced its negotiating strategy."
Welsh Liberal Democrat leader Mark Williams welcomed the ruling, saying it showed the sovereignty of parliament was paramount, and that his party will vote against Article 50.
He added: "The UK government must absolutely respect the devolved settlement of Britain and the right of the devolved parliaments to shape our withdrawal from the EU."
Several Labour MPs have indicated they will disobey any order by party leader Jeremy Corbyn to vote in favour of Article 50.
Pontypridd MP Owen Smith, who backed Remain, suggested there ought to be a fresh referendum on the terms of leaving the EU, fearing the UK was set for the "hardest of hard Brexits" by leaving the single market and the customs union.