The British government's decision to support the additional runway puts an end to rival plans to expand Gatwick airport. A Foster + Partners proposal for a new airport on the Thames estuary had already been ruled out of the running in 2014.
Grimshaw's proposal features a vast terminal hall with an undulating glass roof.
"We are very pleased with the government's announcement regarding the expansion of Heathrow Airport, and believe this is clearly the best way forward for the country," said Grimshaw partner Jolyon Brewis.
"This decision gives credit to the longstanding quality service the airport provides to its passengers, and recognises the transport hub as a key generator in keeping the UK globally connected," he added.
"We look forward to working with Heathrow on the new development, delivering their bold vision and working towards their ambitious and achievable target of the most sustainable airport hub in the world."
UK prime minister Theresa May and London mayor Sadiq Khan had previously opposed plans to expand Heathrow Airport.
"Heathrow will now start the important work to plan, consult and build the vital new runway that the UK needs to compete in the world," said a statement from Heathrow Airport. "We are prepared to move quickly to expand."
It also announced its plans to add 25,000 flights a year from 2021 – ahead of the proposed opening of the third runway in 2025.