Last month, the government asked for further information to help choose between the Heathrow and Arora schemes.
The Department for Transport said Heathrow's own proposal offered the most deliverable option, and the "greatest likelihood" of getting a decision on planning approval within this parliament.
The plan that has been backed will inform the government's review of the Airports National Policy Statement.
Once that is complete, Heathrow is expected to apply for planning permission. The government then hopes for a decision by 2029.
But any company will be able to submit an application to build the new runway and terminals at the site.
Heathrow had set out its plans for expansion in the summer. The whole project, which is expected to cost £49bn, includes:
The Arora Group said it accepted the government's choice, adding it welcomed the decision to leave the option open for other firms to bid for the work.
"It's imperative there is a clear and transparent process for selecting a promoter to ensure it best serves the interests of consumers," the group said.
Transport Secretary Heidi Alexander said Heathrow was the UK's only hub airport that supported trade, tourism and jobs.
"Today is another important step to enable a third runway and build on these benefits, setting the direction for the remainder of our work to get the policy framework in place for airport expansion," she said.
"This will allow a decision on a third runway plan this parliament which meets our key tests including on the environment and economic growth."
The government has already approved a string of other airport expansion plans, including a second runway at Gatwick Airport.
However, the Heathrow plans face opposition from environmental groups, politicians, and local residents.
Tony Bosworth, climate campaigner at Friends of the Earth, said the plan was "virtually the equivalent of bolting an airport the size of Gatwick onto Heathrow".
"That would mean more noise and air pollution for local communities," he said.
"Expanding Heathrow simply isn't compatible with our legally binding climate targets, even if the government meets its hugely optimistic assumptions for emerging technologies, such as sustainable aviation fuels."
The Mayor of London, Sir Sadiq Khan, said he thought the government's backing of a new runway was a mistake.
"I want a better Heathrow, not a bigger one, particularly at a time where Gatwick's going to be expanded, Stansted is going to be expanded, City Airport's going to be expanded, Luton's going to be expanded," he said.
"I'm unclear how you get a new runway at Heathrow and it doesn't cause environmental damage, noise pollution, air pollution being exacerbated."