It means the competing plan for a shorter runway, as proposed by hotel tycoon Surinder Arora, has been rejected.
Heathrow says the project will be 100% privately financed, through higher airline costs, and no taxpayer money will be used to build the runway or the associated infrastructure.
But it will require re-routing the M25 motorway – one of the busiest in the country.
The proposal is still subject to the planning process.
The Department for Transport says selection of the scheme does not represent a final decision on a third runway scheme or design, and any amendments will be subject to consultation and parliamentary scrutiny next year.
Exact details such as the length of the third runway, layout, and associated infrastructure implications will continue to be considered throughout the remainder of the review.
Ms Alexander said: “Heathrow is our only hub airport which supports trade, tourism and hundreds of thousands of jobs, underpinning prosperity not only in the South East but across the UK.
“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. This will allow a decision on a third runway plan this parliament which meets our key tests including on the environment and economic growth.
“We’re acting swiftly and decisively to get this project off the ground so we can realise its transformational potential for passengers, businesses, and our economy sooner.”
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