Judges said the Government failed to take account of its Paris Agreement climate change commitment when setting out its support for the airport expansion in its National Policy Statement.
But the west London hub said it will continue to fight for expansion, saying it is "confident" that it will win in the Supreme Court.
Transport Secretary Grant Shapps has confirmed the Government will not appeal today's Court of Appeal ruling.
He later tweeted: “Airport expansion is core to boosting global connectivity. We also take seriously our commitment to the environment.
“This Govt won’t appeal today’s judgement given our manifesto makes clear any #Heathrow expansion will be industry led.”
Airport expansion is core to boosting global connectivity. We also take seriously our commitment to the environment. This Govt won't appeal today's judgement given our manifesto makes clear any #Heathrow expansion will be industry led.— Rt Hon Grant Shapps MP (@grantshapps)February 27, 2020
A Heathrow spokesperson said: “The Court of Appeal dismissed all appeals against the government - including on “noise” and “air quality” - apart from one which is eminently fixable.
"We will appeal to the Supreme Court on this one issue and are confident that we will be successful. In the meantime, we are ready to work with the Government to fix the issue that the court has raised. "
The airport added: "Expanding Heathrow, Britain’s biggest port and only hub, is essential to achieving the Prime Minister’s vision of Global Britain. We will get it done the right way, without jeopardising the planet’s future. Let’s get Heathrow done.”
When he was London's mayor, Prime Minister Boris Johnson opposed the expansion of the west London and promised to "lie down ... in front of those bulldozers" to stop the runway being built.
But since moving into Number 10 Downing Street Mr Johnson shied away from renewing his pledge, and the 2019 Conservative election manifesto said Heathrow itself must show “the project can be financed and built”.
In the wake of the ruling, Labour's John McDonnell told the Standard: "It was me who invited Boris to lie in front of the bulldozers with me and he said yes.
"Now he is off the hook he can drop this altogether now. Now we can save the planet and local communities.
"It's a terrific verdict for the people of West London. We need green growth and an green industrial revolution that is the way forward."
After the ruling today, Mayor of London Sadiq Khan, who supported the legal action, welcomed the ruling alongside campaigners who want to block the airport from the controversial expansion.
He said: "We won! Today we blocked the Tory government plans to build a third runway at Heathrow Airport. Today's judgment is a major victory for all Londoners who are passionate about tackling the climate emergency and cleaning up our air."
After the hearing, protesters whooped and cheered as campaigners left the Royal Courts of Justice waving copies of the judgment. One group popped a bottle of champagne before handing out plastic cups of the fizz.
John Sauven, executive director of Greenpeace UK, said: "The court has decided that the Airports National Policy Statement is fatally undermined by ignoring climate commitments, but we still need the Government to permanently ground Heathrow's expansion plans."
Aviation Environment Federation deputy director Cait Hewitt said: "This is a huge win for the climate, and leaves Heathrow's third runway plans in tatters.
Caroline Russell, London Assembly member for the Green Party also welcomed the ruling, saying: "I'm wet, soggy and absolutely ecstatic. It's the just decision the right decision. I'm so glad people have seen sense in the court.
"The idea that in a climate emergency we would expand aviation is quite mind-boggling.
"I fear Heathrow will be back but they will eventually realise that business as usual in a climate emergency makes no sense."
Conservative candidate for mayor of London Shaun Bailey said: "Keeping the air clean in London is a priority and another runway was not the answer. It doesn't work on financial or climate grounds."
"A real relief would be for this thing to stop. We don't want to hear any nonsense from them about an appeal."
Tanya Steele, CEO of WWF, said: “This is a victory for the climate and for future generations who will have to live with the impacts of environmentally catastrophic infrastructure projects.
"No plan for net zero emissions, either from the UK Government or from Heathrow itself, can be credible if it includes a third runway.”
At the time, then Transport Secretary Chris Grayling had said a new runway would set a "clear path to our future as a global nation in the post-Brexit world".
Heathrow has said it could open the third runway "between early 2028 and late 2029". Its previous target date was 2026.
The airport said the delay was due to the Civil Aviation Authority, which refused to approve its bid to nearly quadruple the amount it spends before obtaining final planning permission.
Campaigners gathered outside court before the ruling, from 9am.
In their ruling, the judges revealed that the government has already decided not to appeal today’s decision.
“Having seen our judgment in draft the government has not opposed the grant of a remedy.
“Nor has the government sought permission to appeal from our decision to the Supreme Court.”