A number of critics have denounced Mr Bannon as supporting white supremacism.
But Reince Priebus said this was “not the Steve Bannon that I know”, adding he was a “very, very smart person”.
The appointment of Mr Priebus is seen as an attempt by Mr Trump to improve links to the Republican establishment.
Mr Trump himself said he fought the election as the “ultimate outsider”, and it would fall to Mr Priebus, the current chairman of the Republican National Committee (RNC), to act as a conduit both to the party and to Congress.
One of Mr Priebus’s first tasks was to defend the appointment of Mr Bannon, who had stepped aside from his role as chief executive of the Breitbart News Network – a combative conservative site with an anti-establishment agenda – to act as Mr Trump’s election campaign chief.
Adam Jentleson, a spokesman for Senate Democratic leader Harry Reid, said of Mr Bannon: “It is easy to see why the KKK views Trump as their champion when Trump appoints one of the foremost peddlers of White Supremacist themes and rhetoric as his top aide.”
Jonathan Greenblat, of the Anti-Defamation League civil rights group, said: “It is a sad day when a man who presided over the premier website of the ‘alt-right’ – a loose-knit group of white nationalists and unabashed anti-Semites and racists – is slated to be a senior staff member in the ‘people’s house’.”
The Southern Poverty Law Center group said: “Stephen Bannon was the main driver behind Breitbart becoming a white ethno-nationalist propaganda mill… Bannon should go.”
But Mr Priebus told Good Morning America: “I don’t know where they’re coming from… that’s not the Steve Bannon that I know.”
Who is Stephen Bannon?
- Executive chairman of Breitbart news, although stepped aside for the election campaign
- A graduate of Georgetown University and Harvard Business School, former US Navy officer and investment banker at Goldman Sachs
- Conservative documentary film-maker who produced films celebrating Reagan, Sarah Palin and the Tea Party
- Seen as in conflict with the traditional Republican establishment and has been called a racist and right-wing extremist by some members of his own party
Who is Reince Priebus?
- Mr Trump’s campaign adviser was elected as chairman of the Republican National Committee in 2011
- A lawyer and former Wisconsin state treasurer who worked his way up the Wisconsin Republican party to become its chairman
- Commissioned a party review after the 2012 presidential election defeat to Barack Obama aimed at increasing party appeal
Mr Priebus urged people to “look at the person, get to know the person,” saying Mr Bannon “was a force for good on the campaign. I haven’t seen any of these things that people are crying out about. It’s a good team, it works”.
He said that among Mr Trump’s priorities on taking office would be to “get his arms around foreign policy”, to start measures on tax reform and to plan changes to the Affordable Care Act – known as Obamacare.
Mr Priebus said Mr Trump was “very calm and cool and collected – and prepared to lead the American people”.
Unlike Mr Bannon, Mr Priebus’s appointment has won general approval, including from former Obama adviser, David Axelrod.
Mr Priebus is seen as consensus builder who is close to House Speaker Paul Ryan, a fellow Wisconsinite who could be instrumental in steering the new administration’s legislative agenda.
Mr Priebus said: “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.”
Mr Ryan said on Monday that Mr Priebus’s appointment was “a very, very, very good sign of things to come”.
Mr Bannon on Sunday thanked Mr Trump for his appointment, saying: “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 Trump achieve his agenda.”
In a statement released by his campaign, Mr Trump described Mr Priebus and Mr Bannon as “highly qualified leaders who worked well together on our campaign and led us to a historic victory”.
In the president-elect’s first interview, with US broadcaster CBS on Sunday, Mr Trump said he would deport or jail up to three million illegal migrants with criminal links, nominate “pro-life” candidates for the Supreme Court and forgo the president’s $400,000 salary, taking $1 a year instead.
Both houses of Congress are under Republican control.
Donald Trump will take over at the White House on 20 January, when Barack Obama steps down after two terms in office. Mr Trump defeated Hillary Clinton in last week’s presidential vote.