The president said he was recommending a key federal agency “do what needs to be done”, even as he vowed to continue contesting the result.
The General Services Administration (GSA) said it was acknowledging Mr Biden as the “apparent winner”.
Earlier, Mr Biden’s victory in the US state of Michigan was officially certified, in a major blow to Mr Trump.
What did Trump say?
Mr Trump tweeted as the GSA, which is tasked with formally beginning presidential transitions, informed the Biden camp that it would start the process.
Administrator Emily Murphy said she was making $6.3m (£4.7m) in funds available to the president-elect.
While pledging to keep up the “good fight”, the president said: “Nevertheless, in the best interest of our Country, I am recommending that Emily and her team do what needs to be done with regard to initial protocols, and have told my team to do the same.”
…fight, and I believe we will prevail! Nevertheless, in the best interest of our Country, I am recommending that Emily and her team do what needs to be done with regard to initial protocols, and have told my team to do the same.
— Donald J. Trump (@realDonaldTrump) November 23, 2020
Ms Murphy, a Trump appointee, cited “recent developments involving legal challenges and certifications of election results” in her decision to send the letter.
Ms Murphy said she had received no pressure from the White House over the timing of her decision.
“To be clear I did not receive any direction to delay my determination,” said her letter to Mr Biden.
“I did, however, receive threats online, by phone, and by mail directed at my safety, my family, my staff, and even my pets in an effort to coerce me into making this determination prematurely.
“Even in the face of thousands of threats I have remained committed to upholding the law.”
She had faced criticism from both political sides for failing to begin the transition process sooner, usually a routine step between the election and the inauguration.
Ms Murphy missed a deadline on Monday set by Democrats in the House of Representatives to brief lawmakers about the delay.
The Biden team welcomed her letter.
“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.”
What happened in Michigan?
One of two Republicans on the Michigan State Board of Canvassers joined the two Democrats to finalise the result. The other Republican abstained. Mr Biden won the state by more than 150,000 votes.
Abstaining Republican board member Norman Shinkle had suggested delaying the certification over irregularities affecting a few hundred votes in one county.
But his colleague, Republican Aaron Van Langeveld, on Monday said their duty was “simple” and there was no option but certification.
The president’s Republican allies had called for the certification to be delayed for two weeks to audit ballots in a heavily Democratic county.
Mr Trump’s legal team said they would still challenge Michigan’s results.
Adviser Jenna Ellis said certification was “simply a procedural step”. She added: “Americans must be assured that the final results are fair and legitimate.”
But time is running out. On 14 December, Mr Biden’s victory is set to be approved by the US Electoral College.
What about Trump’s other legal challenges?
Mr Trump and his allies have suffered a string of court defeats in key states as they race to challenge the results.
His campaign has reportedly tried to convince Republican state lawmakers to appoint their own electors to vote for him instead of Mr Biden, but to no avail.
In neighbouring Wisconsin, a partial recount is under way by request of the Trump campaign. Election officials have accused Trump supporters of obstructing the state’s recount of votes.
They said observers for Mr Trump were in some cases challenging every single ballot to deliberately slow down proceedings.
In Pennsylvania, a Republican judge on Saturday ruled that the Trump campaign had tried to “disenfranchise almost seven million voters” with no real evidence. The president’s lawyers have appealed now to the Circuit Court of Appeals in Philadelphia.
The president’s other legal efforts in the state have failed to change Mr Biden’s lead of some 80,000 votes.
The Trump campaign has also called for another recount in Georgia, after an earlier recount by hand confirmed Mr Biden’s win in the state.