In this role, he will set the tone for the new White House and act as a conduit to Congress and the government.
Bannon, from the conservative Breitbart News Network, will serve as Mr Donald Trump's chief strategist.
Mr Stephen Bannon stepped aside temporarily as Breitbart's executive chairman to act as Mr Donald Trump 's campaign chief.
Mr Steve Bannon stepped aside temporarily as Breitbart's executive chairman to act as Mr Donald Trump's campaign chief.
The Republican candidate defeated Democrat Mrs. Hillary in Tuesday's presidential vote, in a result which shocked many, who had expected Mrs Clinton to win following favourable opinion polls.
Mr Donald Trump is due to take over at the White House on 20 January, when Mr Obama steps down after two terms in office.
Meanwhile in the president-elect Donald Trump's first interview, with US broadcaster CBS, Mr Donald Trump said:
I am thrilled to have my very successful team continue with me in leading our country," Mr Donald Trump said in a statement released by his campaign.
Steve Bannon and Reince Priebus are highly qualified leaders who worked well together on our campaign and led us to a historic victory. Now I will have them both with me in the White House as we work to make America great again.
Mr Reince Priebus, 44, acted as a bridge between Mr Trump and the Republican party establishment during the campaign.
He is close to House Speaker Paul Ryan, a fellow Wisconsinite, who could be instrumental in steering the new administration's legislative agenda.
The man who built his campaign railing against The Establishment has chosen the chair of the Republican National Committee to be his chief of staff. It doesn't get much more establishment than Reince Preibus.
If there are clues to be gleaned from Mr Trump's first personnel decisions as president, it's that he's opting for a veteran party hand to manage the White House - although he's keeping an outsider devil on his shoulder in senior adviser Steve Bannon, former head of the bomb-throwing Breitbart News.
Bringing Mr Reince Priebus and Mr Steve Bannon under the same roof should create some interesting tension. If correctly harnessed, the energy could provide drive to the nascent Donald Trump administration. If things go wrong, it could tear the place apart.
Regardless of how it works out, Mr Reince Priebus's elevation to this powerful position represents the culmination of a winning gamble for the Wisconsinite. While many in his party were urging him to abandon Mr Donald Trump whenever his candidacy appeared on the verge of foundering, he committed - quietly, behind the scenes - to righting the ship.
It often didn't seem possible, but he succeeded and now he has a White House office to show for it.
Elected chairman of the Republican National Committee in 2011, Mr Reince Priebus has acted as the party's spokesman and chief fundraiser, helping candidates running for re-election.
He said it was truly an honour to join Mr Donald Trump in the White House as chief of staff.
I am very grateful to the President-elect for this opportunity to serve him and this nation as we work to create an economy that works for everyone, secure our borders, repeal and replace Obamacare and destroy radical Islamic terrorism, he added.
Correspondents say one of the big challenges of the new administration will be reconciling Mr Donald Trump with the mainstream Republican party, where sharp divisions emerged during the primaries.
Both houses of Congress are under Republican control.
During the election race, Mr Steve Bannon, 62, saw it as his aim to bolster the business-like approach of Mr Donald Trump's campaign.
A former naval officer, investment banker and Hollywood producer, Mr Steve Bannon took over at Breitbart in 2012, when he promised to make it the Huffington Post of the right.
Breitbart is the most-read conservative news website in the United States of America, set up to challenge the power of mainstream media.
I want to thank President-elect Donald Trump for the opportunity to work with Reince in driving the agenda of the Trump administration, Mr Steve Bannon said on Sunday.
We had a very successful partnership on the campaign, one that led to victory. We will have that same partnership in working to help President-elect Donald Trump achieve his agenda.