The agreement adds to telecoms provider BT's growing portfolio of sports rights as it challenges BSkyB in an area which Sky has long dominated.
BT had already taken over the FA Cup rights for the 2013-14 season after buying out ESPN's UK business. Commercial broadcaster ITV shares the FA Cup rights under the contract that expires next year.
The competition was last broadcast on the BBC in 2008.
"Bringing the FA Cup back to the BBC was something I really wanted to do," said BBC director general Tony Hall.
"I am so pleased that we have achieved it for football fans everywhere.
"There is something very special about big national moments on the BBC and the FA Cup should absolutely be one of them.
"The Olympics, Wimbledon and Glastonbury have shown how our audiences love it when we put the full weight of all our services on TV, radio and digital behind covering these events in depth and in ways no-one else can.
"Working closely with the FA I believe we will change the way we view the FA Cup forever."
ITV, meanwhile, have responded by confirming that they will have exclusive rights for all England internationals from next year.
The channel had already bought the rights to World Cup and European Championship qualifiers from UEFA, but have now added all home and away friendly internationals in a separate deal with the FA.
The deal also runs until 2018.