Mr Bannon, a right-wing ideologue who co-founded the Breitbart News website, pronounced himself "free" and said he now had "my hands back on my weapons" at the outlet.
He and Mr Trump's new chief of staff, John Kelly, agreed he would leave the White House on Friday. Mr Bannon said he had given his resignation earlier in the month, though it was also reported that Mr Trump had decided to let him go.
He told the Weekly Standard, a conservative opinion magazine: "The Trump presidency that we fought for, and won, is over.
"We still have a huge movement, and we will make something of this Trump presidency. But that presidency is over. It’ll be something else. And there’ll be all kinds of fights, and there’ll be good days and bad days, but that presidency is over."
During Mr Trump's "America first" campaign and his early days in the Oval Office, Mr Bannon was considered a key influencer and was even awarded a place on the National Security Council's principals committee, the top interagency group overseeing national security – though he was later removed.
His nationalist stances on issues like immigration, trade and society were reflected in Mr Trump's speeches and policies, and are widely seen as having drawn together the New Yorker's support base after he joined the struggling campaign as chief executive last August.
But once in power, he was forced to compete for influence with other advisers including members of Mr Trump's family. Ivanka Trump and her husband Jared Kushner – seen as being able to soften the President's tone and actions – both occupy official White House posts.
Mr Bannon said he believed the Republican Party would now begin to impose a moderating influence on the President, despite his public clashes with senior figures like Senate majority leader Mitch McConnell.
He added: "The path forward on things like economic nationalism and immigration, and his ability to kind of move freely... I just think his ability to get anything done – particularly the bigger things, like the wall, the bigger, broader things that we fought for, it’s just gonna be that much harder."
And he vowed to "crush the opposition" – liberals, the Washington and Republican "establishments" – by using the influence and reach of Breitbart.
He said: "I feel jacked up. I built a f***ing machine at Breitbart. And now I’m about to go back, knowing what I know, and we’re about to rev that machine up."
Breitbart colleague Joel Pollak suggested the site would now go to "war" with the Trump White House following a perceived shift by the President away from the values the outlet espouses.
Mr Bannon's departure came after a liberal magazine published an interview in which he detailed his behind-the-scenes battles with opponents in the administration and appeared to contradict Mr Trump's public rhetoric on North Korea.
He told The American Prospect he was "fighting" internal opposition to his belief the US was locked in "an economic war with China".
He claimed officials at the Treasury and the National Economic Council were "wetting themselves" over his plans to address the fact that "they're crushing us". State and Defence Department staff felt similarly because they wanted China's help to reign in Pyongyang, he said.
Mr Trump has repeatedly and publicly threatened military action against North Korea if it continues to menace the US or its allies.
But Mr Bannon told the Prospect: "Until somebody solves the part of the equation that shows me that 10 million people in Seoul don’t die in the first 30 minutes from conventional weapons, I don’t know what you’re talking about, there’s no military solution here, they got us."