The current broadcasters are ITV and BT Sport through its acquisition of Disney’s UK ESPN business earlier this year.
The BBC’s coverage will be led by Gary Lineker and will be available on TV, online, and via mobile phones and tablets.
Jake Humphrey will lead BT Sport’s coverage from the new channel’s studios in the Queen Elizabeth Park, Stratford.
The Football Association announced the deal at Wembley Stadium on Wednesday.
FA Chairman and former BBC Director-General, Greg Dyke, said: “I would like to thank everyone who expressed an interest in these FA Cup broadcast rights and congratulate the BBC and BT Sport on their successful tenders.
“The BBC is obviously an organisation I know very well but I don’t think anyone would dispute that their name goes hand in hand with some of the most famous FA Cup moments of yesteryear.
“We’re delighted that the competition is back on their channels, not just through television but with significant online presence as well.
“BT Sport might be new to television but they’re a major corporation and have been a part of everyday British life for a number of decades.”
BBC bosses say their coverage will be supported “with a raft of support programming and a top name presentation team” and promises the FA Cup |”will become an integral part of the BBC One schedule.”
BBC Director-General Tony Hall said, “Bringing the FA Cup back to the BBC was something I really wanted to do and I am so pleased that we have achieved it for football fans everywhere. “
Gavin Patterson, CEO, BT Retail: said: “BT Sport made a determined joint bid to retain The FA Cup rights because we believe it is one of the truly great club football competitions.
“We are thrilled to bring our viewers the incredible stories of passion, and victories against all odds, that pepper the history of the competition. These include Leeds United being humbled by Second Division Sunderland in the 1973 final, right up to May this year, when unfancied Wigan Athletic defeated Manchester City at Wembley.”