Activating the CRAF will help military aircraft to focus on operations in and out of Kabul, the statement added.
"These civil reserve flights will help facilitate the safe transport [of evacuees] to third countries," Mr Biden said on Sunday. "None will be landing in Kabul."
The president said a series of "processing stations" had been established in more than two dozen countries, where evacuees would be screened and cleared.
"We will welcome these Afghans who have helped America to their new homes - because that's who we are," Mr Biden added.
The news comes as White House national security adviser Jake Sullivan announced there were still several thousand US citizens thought to be in Afghanistan.
Speaking to CNN's State of the Union programme on Sunday, Mr Sullivan said he could not give a precise number, but noted that work was continuing to evacuate them.
Mr Sullivan also described the threat of attacks by the Islamic State group against the airport as "real" and "acute".
image captionThe civilian aircraft will help the US military with their evacuations
Taliban official Amir Khan Mutaqi meanwhile has blamed the US for the "evacuation drama" at the airport.
On Sunday, Mr Mutaqi also addressed potential dissatisfaction within Taliban ranks, arguing that "certain decisions are made in the long-term interest" of the movement and its role in Afghanistan.
He said that the Taliban were in talks with "all factions" to reach an agreement on a future government.
Johnson calls G7 meeting
The Taliban's swift takeover of Afghanistan has shocked its people and the world.
UK Prime Minister Boris Johnson has called an urgent meeting for Tuesday of the leaders of the G7 - the world's seven largest so-called advanced economies.
"It is vital that the international community works together to ensure safe evacuations, prevent a humanitarian crisis and support the Afghan people to secure the gains of the last 20 years," he tweeted.
Mr Johnson is expected to urge Mr Biden to keep American forces in Afghanistan beyond 31 August in order to allow evacuation flights to continue.
Former UK Prime Minister Tony Blair, meanwhile, has harshly criticised the US withdrawal from Afghanistan.
"You look round the world and the only people really cheering this decision are the people hostile to Western interests," he said.
The US plans a full withdrawal of its troops from Afghanistan on 31 August.
Several allied nations - including the UK - are calling for this deadline to be extended amid the ongoing evacuation, as US soldiers are currently controlling the airport.
By BBC