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 GSA announcement means the president-elect now has access to top security briefings, office space and government officials as he prepares to take office on 20 January.
His transition website has now changed to a US government domain. He will begin announcing the people he wants in the top jobs in his cabinet on Tuesday.
What Biden's new foreign policy team tells us
Biden to pick 'first US female treasury secretary'
John Kerry named as Biden's climate tsar
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How did the Biden team respond?
It 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," a team statement said.
Earlier, Mr Biden unveiled his nominations for his foreign policy and national security team, consisting of old colleagues from his years in the Obama administration.
He is set to 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.
The list of selections came ahead of a formal announcement Tuesday. Most of the appointments will require Senate confirmation.
Has Trump conceded?
Mr Trump tweeted as the GSA, which is tasked with formally beginning presidential changeovers, informed the Biden camp that it would start the transition process.
GSA administrator Emily Murphy said she was making $6.3m (£4.7m) in funds available to the president-elect.
Mr Trump said: "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."
He did not concede, however, and went on to repeat unsubstantiated claims of corruption, pledging to keep up the "good fight". It is worth noting that Mr Trump does not have to concede for Mr Biden to be sworn in as the 46th US president.
Ms Murphy, a Trump appointee, cited "recent developments involving legal challenges and certifications of election results" in her decision to start the transition process.
She said she had received no pressure from the White House over the timing of the move.
"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," Ms Murphy said in a letter to Mr Biden.
"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.
The three people chosen by US President-elect Joe Biden to lead his foreign policy team will bring decades of diplomatic experience to the White House, but they also face criticism from those who object to the baggage that comes along with years of serving the US government.
Here's a little on who they are and then what the experts say about the message Mr Biden is sending with his choice of top US diplomats.
Who are they?
The three people who will be advising Mr Biden on issues beyond America's borders are not well known beyond Washington. Antony Blinken, Linda Thomas-Greenfield and Jake Sullivan are all alumni of the Barack Obama White House, and are considered Biden loyalists and foreign policy centrists.
Mr Blinken, 58 - whose work with Mr Biden goes back nearly 20 years - has been picked to serve as secretary of state, the nation's top diplomat.
Linda Thomas-Greenfield, one of the most high-profile black female US diplomats who worked for years on African affairs, has been nominated to serve as US Ambassador to the United Nations.
Jake Sullivan is a former state department official and Hillary Clinton aide who played a key role in negotiating the 2015 Iranian nuclear deal. He served as Mr Biden's national security adviser when he was vice-president.
How will Biden change US foreign policy?
The insiders
Mr Blinken has worked with Mr Biden for nearly 20 years, dating back to Mr Biden's tenure on the Senate Foreign Relations Committee. Some critics have blamed him for Mr Biden's vote to invade Iraq in 2003.
That personal connection will serve the team well, say several former US diplomats, but it could also mean a lack of diversity in opinion.
And their wealth of Washington experience as foreign policy veterans will not endear them to all. It does, however, mark a break with Trump's war on the so-called "deep state" - individuals in government he regarded as working against his own agenda.
This is not a team of rivals. While they have a history of working well together, they will also bring complementary perspectives to the Situation Room, with hands-on experience in global conflicts. The value of strategic coherence will outweigh any potential for groupthink. They are not card-carrying members of a deep state, but will help lead a more effective American state focused on shared interests and values. They are pragmatists, not ideologues. PJ Crowley, former US assistant secretary of state
The appointments represent a return to normalcy and an end to an erratic approach to policy. But the question is whether a return to normalcy is enough. This team has a belief in American supremacy which could lead them to recklessness in their use of American military power. Andrew Bacevich, president of the Quincy Institute for Responsible Statecraft
John Kerry named as Biden's 'climate tsar'
Biden to pick 'first US female treasury secretary'
As a longtime foreign service officer, Mrs Thomas-Greenfield in particular, represents the "deep state" that Donald Trump was so dismissive of. For Donald Trump, foreign service officers were the foot soldiers of the liberal internationalism that he wanted to take down. This is partly the reason for why morale is so low at the State Department right now. Charles Kupchan, former member of the White House National Security Council
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Global alliances
Top of the agenda for the Biden team will be re-joining organisations, alliances and treaties that Mr Trump worked to weaken or dissolve over the past four years.
They will be tasked with bringing the US back into the Paris climate agreement and keeping the US in the World Health Organization. They will also seek to revise the Iran nuclear deal, bolster ties with Nato and seek trade agreements as a means to counter China's rising influence.
One of the reasons that Trump was elected was because a large number of Americans judged American foreign policy as having failed. The question is whether this team has learned enough to bring about the changes that we actually need including reducing the military budget and greater restraint in the use of military forces. Andrew Bacevich
Donald Trump never really built a national security team. Instead, he played the Lone Ranger while surrounding himself with independent contractors he could hire and fire by tweet. All three of these officials have served at the Department of State and will place renewed value on international diplomacy and collaboration to solve urgent challenges from the global pandemic to climate change to the China challenge. PJ Crowley
A big challenge is going to be persuading foreign audiences, even partners and allies, that the US is credible in terms of its staying power - that "America is back" as the president-elect has said. This will be the second 180-degree shift in foreign policy in the past four years. The incoming administration has a determination to restore the frayed alliances that the US has relied on for more than seven decades. But they have their work cut out for them. Stewart Patrick, Council on Foreign Relations director of the International Institutions and Global Governance programme
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Return of the experts?
Most experts agree that the team Mr Biden is putting in place is more experienced than those appointed by Mr Trump, such as his first Secretary of State Rex Tillerson, who was a former Exxon Mobil executive.
But Mr Trump's legacy at the State Department - which has seen scores of disillusioned diplomats retire early under Mr Tillerson and his replacement Mike Pompeo - run deep and cannot be swiftly overdone overnight.
There has been a significant departure of expertise and experience under both Tillerson and Pompeo. The foreign service bureaucracy has felt generally out of the loop. Normally policy comes up from below, and is filtered through several agencies before reaching the president. That has not been happening under Trump. Charles Kupchan
They have a consistent world view. They are strong believers in American leadership and international alliances. All three have served at the Department of State and will place renewed value on international diplomacy and collaboration to solve urgent challenges from the global pandemic to climate change to the China challenge. PJ Crowley
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Who is Janet Yellen?
The daughter of a family doctor and an elementary school teacher, she grew up in New York City. She earned her degree in economics from Brown University, before going on to complete a PhD at Yale.
In addition to her work at the Fed and in government, she was a professor at University of California, Berkeley.
She is married to George Akerlof, a Nobel Prize-winning economist whom she met while working as a researcher at the Fed in the 1970s. They have one son, who is also an economics professor.
Her climb to the top of the economics profession also made her a feminist icon in the economics world.
When she left the Fed in 2018, many paid tribute to her leadership by imitating her signature look of a blazer with a popped collar.
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How does she fit into the US political scene?
Ms Yellen has a long history of working in Washington. Before Mr Obama named her to lead the Fed in 2014, she had served as one of its board members for a decade, including four years as vice-chair.
At the Treasury, she is likely to be charged with guiding the Biden administration's economic response to the pandemic, which has prompted the worst economic contraction the US has seen in decades and cast more than 10 million people out of work.
The office also oversees tax policy, public debt and international finance and sanctions.
Ms Yellen is seen as someone able to satisfy both progressive and centrist members of Mr Biden's Democratic party. Her nomination to lead the Fed in 2014 won support from some Republicans.
"2 cheers for Janet Yellen at Treasury," liberal public policy professor and former Labor Secretary Robert Reich wrote on Twitter.
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While she is not as left-wing as some of the names rumoured to be under consideration, such as Senator Elizabeth Warren, she still "understands the huge toll stagnant wages, systemic racism, and widening inequality have taken on our economy and society", he said.
Ms Warren also added her congratulations, focusing on Ms Yellen's record as bank regulator. One of her final moves as Fed chair was issuing an order limiting Wells Fargo's ability to grow, in response to a fake accounts scandal at the bank.
"Janet Yellen would be an outstanding choice for Treasury Secretary," Ms Warren said. "She is smart, tough, and principled."
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