The president said the federal agency overseeing the handover must "do what needs to be done", even as he vowed to keep contesting his election defeat.
The General Services Administration (GSA) said it was acknowledging Mr Biden as the "apparent winner".
It came as Mr Biden's victory in the state of Michigan was officially certified, a major blow to Mr Trump.
The Biden team welcomed the start of the transition process as the Democratic president-elect gears up to be sworn in on 20 January.
"Today's decision is a needed step to begin tackling the challenges facing our nation, including getting the pandemic under control and our economy back on track," said its statement.
"This final decision is a definitive administrative action to formally begin the transition process with federal agencies."
Earlier on Monday, Mr Biden unveiled a foreign policy and national security team consisting of old colleagues from his years in the Obama administration.
He will appoint Anthony Blinken as secretary of state and John Kerry as climate envoy, while Janet Yellen is tipped to be the first female US treasury secretary.