The BBC said it would "reluctantly" end its TV contract three years early in order to make the savings. BBC Sport must make £35m of budget cuts to contribute to the overall £150m annual savings needed.
"The current financial position of the BBC means some tough and unwanted choices have to be made," said BBC sport director Barbara Slater. "There are no easy solutions; all of the options available would be unpopular with audiences."
Read more: Which Bond is the biggest gambler?
Radio rights have been extended until 2021, however.
Channel 4 has bagged the new broadcasting rights and will begin airing the Grand Prix from next season along with Sky, replacing the BBC.
The three-year deal with Formula 1 includes 10 live races per season and highlights of all 21 races. In an unusaul move for a comemrcial broadcaster, no adverts will be shown during any of the live races, Channel 4 said.
“I am sorry that the BBC could not comply with their contract but I am happy that we now have a broadcaster that can broadcast Formula events without commercial intervals during the race," said F1 chief Bernie Eccelstone.
“I am confident that Channel 4 will achieve not only how the BBC carried out the broadcast in the past but also with a new approach as the World and Formula 1 have moved on.”
The value of the new deal has not been disclosed. Details of the production, presenters and races that will be shown will be revealed in due course.
The BBC had saved £150m in 2011 by splitting the rights with Sky - the first time the sport had not been completely free-to-air. At the time, the BBC said the compromise would allow it to keep the sporting Crown Jewel of Wimbledon.
The deal is a coup for Channel 4 which has been increasing its sports output following success with the 2012 Paralympics.
However, the future of the broadcaster, which is also a publicly owned like the BBC but funded entirely by advertising, is also left with a question mark above it. The government is said to be considering privatisation.
Channel 4 said the purchase of Formula 1 rights would not affect its investment in original programming.