The cabinet signed off the plans after they were approved by the government's economic sub-committee, which is chaired by Prime Minister Theresa May.
Transport Secretary Chris Grayling announced £2.6bn in compensation for residents and noise abatement measures.
Environmental groups oppose the plan, which Mr Grayling says will only happen if air quality commitments are met.
"The time for action is now," Mr Grayling told MPs, who will be asked to vote on the expansion plan by 11 July.
He insisted the decision was being taken in the national interest and would benefit the whole of the UK - with 15% of new landing slots at the airport "facilitating" regional connectivity.
He said the £14bn runway, which could be completed by 2026, would be funded entirely privately - but MPs warned that taxpayers would end up footing the bill for billions in road improvements and other upgrades and warned that the UK's carbon emission targets would be threatened by the increase in traffic around the enlarged airport.
A decision - finally?
The debate on expanding Heathrow has been going on for nearly 20 years.
The last Labour government backed the idea, and won a vote on it in 2009, but that plan was scrapped - and the idea of expansion put on hold for five years - by the Conservative-Lib Dem coalition formed after the 2010 election.
But the idea of expansion was resurrected and has been subsequently backed by the Conservatives.
Ministers approved a draft national airports policy statement in October but Parliament has yet to give its approval for detailed planning to begin.
Opponents have threatened a legal challenge while Foreign Secretary Boris Johnson, who is MP for Uxbridge and South Ruislip in west London, has vowed to "lie down in front of bulldozers" to prevent it.
The BBC's assistant political editor Norman Smith said ministers whose constituencies would be directly affected might be given a "get out of a jail card" - by being allowed to miss the vote or even vote against.
No 10 said Mrs May has written to ministers to say those with long-standing objections to a third runway will be permitted to restate their views at a local level, but not to campaign actively against the decision.
Campaigners argue that a new runway will breach the UK's legal limits on air pollution and increase noise pollution with an extra 700 planes a day.
It will result in huge disruption to residents of nearby villages, such as Longford, Harmondsworth and Sipson, with hundreds of homes likely to be knocked down.
Robert Barnstone, from the No 3rd Runway Coalition, told the BBC it was a "disappointing" day and the government was "failing people and failing the environment as well".
Former Transport Secretary Justine Greening, who backs expanding Gatwick instead, suggested the idea of Heathrow as a national hub airport was outdated and the focus should be on improving regional capacity.
"We are now moving to point-to-point travel," she told BBC Radio 4's Today. "Why should people who are living in Newcastle spend hours travelling down to London, then fly out somewhere else?
"There is nothing national about this national policy statement. It is just a runway in Heathrow."
And Tory MP Zac Goldsmith, who resigned his Richmond Park seat in 2016 over the issue and subsequently lost a by-election, said for many people "this doesn't just look like a blank cheque being given by this government to a foreign-owned multinational, it looks like a whole book of cheques signed by our constituents".
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