There will also be job cuts at 5Live as well as a review of the number of presenters the BBC has and how they work.
The corporation said it would reorganize its newsroom along a "story-led" model where staff will be assigned to stories and not attached to individual programs.
"We need to reshape BBC News for the next decade in a way which saves substantial amounts of money".
The spending cuts will mean a reduction in the number of films produced by Newsnight leading to job losses on the BBC Two programme. As part of its plan, it said more BBC journalists would be based outside the capital in future.
Like many media organisations, the BBC, which is the world's largest news broadcaster, is battling new ways to win audiences, as news and entertainment consumer habits change.
"If we adapt we can continue to be the most important news organisation in the world".
"I believe that we have a vital role to play locally, nationally and internationally".
"In fact, we are fundamental to contributing to a healthy democracy in the United Kingdom and around the world", she added.
The cuts also come amid payouts to some female staff, with radio presenter Sarah Montague getting a £400,000 settlement and Samira Ahmed winning an employment tribunal in a dispute over equal pay.
Confirming the decision, Ms Unsworth said: "Linear television viewing is declining, and as we progress with our £80 million savings target, it is no longer cost-effective to continue producing the programme on television".
Director general Tony Hall has announced plans to resign over the summer, while the broadcaster is bracing for tough talks with the government, which is threatening to decriminalize non-payment of the licence fee - a move that the BBC predicts would give it £200M ($260M) less to spend on content.
A standard licence costs each British household just over £154 ($202, 182 euros) a year, and is legally needed to watch any live television. "Ever. We were asked to grow our digital audience - we did - our digi figures are huge (our successful digital figures appear to be an inconvenience to those making the decisions)".