Trump's choices came as he reaffirmed plans to immediately deport or jail as many as three million undocumented immigrants, in his first television interview since his election.
Millions were expected to tune in to Trump's full interview with CBS's "60 Minutes" for clues on how the populist billionaire will govern, and how far he intends to convert his strident slogans into hard and fast policy.
Trump's two sides -- the practical dealmaker and the anti-establishment provocateur -- were on display in his appointments of Bannon, the CEO of right-wing, conspiracy-mongering Breitbart website, and Priebus, a seasoned Republican operative with close ties to House Speaker Paul Ryan.
"Steve and Reince 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."
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The appointments to Trump's inner circle are seen as key to setting the tone of an administration led by a 70-year-old political novice
Since Tuesday's shock election triumph, Trump had appeared to tone down his rhetoric, notably suggesting he might be willing to reconsider a pledge to scrap President Barack Obama's signature health reform.
But he made clear in excerpts of Sunday's interview that he still intended to crack down on the undocumented, focusing on people with criminal records.
"What we are going to do is get the people that are criminal and have criminal records, gang members, drug dealers, where a lot of these people -- probably two million, it could be even three million -- we are getting them out of our country or we are going to incarcerate," Trump said.
Experts estimate there are as many as 11 million foreigners living illegally in the United States, many of them Central and Southern Americans long-established in the country.
Trump also said he stood by his pledge to build a wall on the Mexican border -- although he said it could include some fencing.
His stance stood in opposition with comments by House Speaker Paul Ryan, who said Sunday the focus under a Trump administration would be on securing the border, not rounding up immigrants.
"We are not planning on erecting a deportation force. Donald Trump's not planning on that," Ryan told CNN.
But Newt Gingrich, a member of Trump's inner circle, said he would advise the president-elect to "swing for the fences."
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"This is a city which if you don't shove it as hard as you can while you have momentum, it will just surround you. I mean, the swamp doesn't want to be drained. And the swamp will just suck you in if you let it," he said on CBS's Face the Nation.
Since his election on the back of an incendiary anti-immigrant campaign, thousands of demonstrators have taken to the streets daily, worried that Trump will put his rhetoric into practice.
The billionaire's Trump Tower residence in Manhattan was picketed for four straight days, with similar protests across the United States.
The tower has been a hive of activity as the real estate mogul huddles with his transition team, tackling the colossal task of filling cabinet posts and hundreds of top government jobs ahead of his January 20 inauguration.
Democrats have been despondent in the wake of an election they were certain of winning, but vowed Sunday to oppose Trump -- despite now being locked out of power not only in the House and Senate, but now in the White House as well.
"Our job now is to hold him accountable," said Bernie Sanders, who lost to Hillary Clinton in the Democratic nomination race.
"If Mr. Trump has the courage to take on Wall Street, to take on the drug companies, to try to work forward, go forward to create a better life for working people, we will work with him, issue by issue.
"But if his presidency is going to be about discrimination, if it's going to be about scapegoating immigrants or scapegoating African-Americans or Muslims, we will oppose him vigorously," Sanders declared.
An ABC News/Washington Post poll out Sunday shows that 74 percent of Americans accept Trump's election as legitimate, but that number fell to 58 percent among supporters of his defeated rival Clinton.
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Trump has shown some willingness to soften his more strident positions. His U-turn on Obamacare -- he now says he may simply amend a law he once branded a "disaster" on the stump -- was prompted by his White House meeting with the outgoing president earlier this week.
He told CBS and The Wall Street Journal he may maintain some of the program's more popular elements, such as a ban on insurance companies denying coverage because of so-called pre-existing health conditions.
Asked by the paper whether he would, as threatened, name a special prosecutor to investigate Clinton over her use of a private email server as secretary of state, Trump deflected, saying his priorities were "health care, jobs, border control, tax reform."
Not long ago Trump was leading crowds in chants of "Lock her up!"
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