Costa has almost 4,000 stores across 32 countries.
Speaking on the BBC's Today programme, Ms Brittain explained that Coca-Cola wanted to buy Costa because "they want the coffee product, they have no coffee in their range".
Whitbread acquired Costa in 1995 for 19 million pounds when it had only 39 shops.
This story will be updated. It has since grown from 39 outlets to more than 2,400 coffee shops in the United Kingdom and 1,400 in 31 overseas markets.
Whitbread will now focus on its Premier Inn hotel chain.
While Costa does not compete in the US market, it is the U.K.'s leading coffee chain and operates several hundred outlets in China.
Whitbread shares rose as much as 19% as the sale gives the United Kingdom company a quick way out.
"Hot beverages is one of the few remaining segments of the total beverage landscape where Coca-Cola does not have a global brand", Coca-Cola Chief Executive Officer James Quincey said in the statement.
Proceeds of the sale will be used to pay down debt and boost the pension fund.
The deal, worth £3.9bn, has been unanimously approved by Whitbread shareholders.
Shares in Whibread surged nearly 20% on the news, adding more than £1bn to the group's stock market value, as analysts said the price was substantially higher than expected.
The company said that "a significant majority" of the...
Costa is owned by Whitbread, the British multinational hospitality company.
Whenever Alison Brittain, the chief executive of Whitbread, was asked in the past whether she'd consider spinning off Costa, she'd say it was more complicated than people realised.
"Its global reach should turbo-charge growth in the years to come, and hot drinks are one of the few areas of the wider beverages sector where the soft drinks giant doesn't have a killer brand". The company will return most of the proceeds to its shareholders, it said. "I'd like to welcome the team to Coca-Cola and look forward to working with them".