The changes also drew sharp criticism from Obama's national security adviser, Susan Rice, who said in a Twitter message that the moves were "stone cold crazy".
"Who needs military advice or (intelligence) to make policy on [Isis], Syria, Afghanistan, DPRK?" she chided.
Rice also criticised aspects of the order that would let Vice-President Mike Pence chair meetings of the council in lieu of the President, and reduced the role of the US ambassador to the United Nations.
Spicer called Rice's criticism "clearly inappropriate" and said "the President gets plenty of information from the chairman of the Joint Chiefs of Staff."
The changes were intended to "modernise the National Security Council so that it is less bureaucratic and more focused on providing the President with the intelligence he needs," Spicer said.
The White House spokesman also defended Bannon's inclusion in the group, saying the aide was a "former naval officer" with a "tremendous understanding of the world and the geopolitical landscape that we have now".
Bannon has become one of the President's most trusted, and most contentious, advisers for his ability to channel the populist and nationalist sentiment that helped propel Trump to the Oval Office. In his job at Breitbart, Bannon called the website a platform for the "alt-right," a brand of conservatism known for frequent inflammatory statements on race and other issues.
Robert Gates, Obama's Defence Secretary and a veteran of the NSC and CIA, said in an interview on the same ABC broadcast that while he wasn't concerned about Bannon's inclusion, he did believe pushing the DNI and military out of meetings was a "big mistake".
"They both bring a perspective and judgement and experience to bear that every president, whether they like it or not, finds useful," Gates said.
White House Chief of Staff Reince Priebus downplayed the move, saying on NBC that the intelligence director and chairman of the joint chiefs would be "included as attendees anytime they want to be included".
Still, the moves raise questions about whether the role of the DNI, which was created to better coordinate intelligence agencies after the September 11 attacks, would diminish under the Trump Administration.
Michael Flynn, Trump's national security adviser, resigned in 2014 from his job as director of the Defence Intelligence Agency amid pressure from James Clapper, who was director of national intelligence under Obama. Flynn also clashed with others in the Obama Administration over his management style and priorities.
Spicer said that "you've got a leader in General Flynn who understands the intelligence process and the reforms that are needed probably better than anybody else".
But Flynn's role as a White House gatekeeper on national security also has been questioned because - unlike Defence Secretary James Mattis and other officials in the new Administration - he shares Trump's optimism about efforts to forge better relations with Russian President Vladimir Putin.
