Tony Hall, head of the Royal Opera House (ROH), was handed the £450,000-a-year role after being directly approached by the BBC Trust days after George Entwistle resigned from the job.
The appointment of Lord Hall, a former BBC news executive who has been chief executive of the ROH since 2001, has been hailed as providing the corporation with some leadership.
Tim Davie will remain as the acting director-general until Lord Hall is able to take up the post next March.
BBC Trust chairman Lord Patten said Lord Hall was the right person to lead the BBC as it takes "a long, hard look at the way it operates and puts in place the changes required to ensure it lives up to the standards the public expects".
The new director-general, the only candidate contacted by the Trust, said: "It's been a really tough few weeks for this organisation and I know we can get through it by listening patiently and thinking carefully about what to do next.
"I care passionately about the BBC, about what it can do, its programme-makers and the impact we have in all sorts of different ways.
"It's one of those extraordinary organisations which is an absolutely essential part of Britain, of the UK, of who we are. But it also has this incredible impact around the world too.
"I know that with the right creative team in place, working off each other, sparking off each other, giving each other ideas, you can do extraordinary creative things and I want to build a world-class team for this world-class organisation."
The 61-year-old, made a cross-bench peer in 2010, did not take questions about his appointment, and instead went to meet BBC staff. He has retained his interests in broadcasting as deputy chairman of Channel 4.
Culture Secretary Maria Miller praised his appointment and said: "He has a very strong track record in successfully leading iconic organisations."
Broadcaster David Dimbleby said: "I think it's a very good choice and a great relief for those of us who work for the BBC."
Lord Hall will have to rebuild the BBC's battered reputation after weeks of difficulties precipitated by the Jimmy Savile abuse scandal and a report on BBC2's Newsnight programme, which mistakenly implicated Lord McAlpine in child abuse. That blunder led to Mr Entwistle quitting his post and resulted in the BBC settling with Lord McAlpine for £185,000.
A number of inquiries into the Savile fallout are under way.
Lord Hall's appointment came after BBC Trustee Anthony Fry defended the payment of a year's salary of £450,000 to Mr Entwistle despite him quitting on November 11, 54 days after he took office, double what he would normally have been entitled.
Mr Fry told the House of Commons Public Accounts Committee it was the right thing to do to avoid a legal wrangle.
The direct approach for Lord Hall's recruitment is in marked contrast to the team of headhunters involved in drawing up the list of candidates which resulted in Mr Entwistle, a BBC insider, being given the post.
Lord Patten said: "I believe the approach we have taken is ultimately in the interests of the BBC and, most importantly, licence fee-payers as we have got the best candidate and he will help the organisation quickly get back on an even keel."
Lord Hall, who began as a news trainee with the BBC 39 years ago, is thought to have been in the running for the director-general post when Greg Dyke was appointed in 1999. He was not an applicant when the position was vacated by Mark Thompson earlier this year.
The former head of BBC news and current affairs from 1996 to 2001 said he wanted to lead a "world-class BBC".
He said: "I'm committed to making this a place where creative people, the best and the brightest, want to work."
Question Time host Dimbleby, who recently complained the BBC is "still over-managed and the management still speak gobbledegook", said: "I think most people will be thrilled at this choice and will also get the leadership that is needed from somebody who is a creative man and a good administrator – and a calm man in a time of crisis."
Lord Hall is credited with turning around the fortunes and public perception of the once troubled Royal Opera House.
Simon Robey, who chairs the ROH's board of trustees, said: "I can think of nobody better able to bring stability back to the BBC. They see, as we do, his qualities of leadership and his depth of relevant experience."