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.
What has the Republican Party been saying?
Although most of Mr Trump's fellow Republicans have united behind his refusal to accept defeat until now, some have increasingly been breaking rank.
Tennessee Senator Lamar Alexander, who is retiring, said the president should "put the country first" and help Mr Biden succeed.
"When you are in public life, people remember the last thing you do."
West Virginia Senator Shelley Moore Capito said "at some point, the 2020 election must end".
More than 160 business leaders had also urged Ms Murphy in an open letter to immediately recognise Mr Biden as president-elect.
"Withholding resources and vital information from an incoming administration puts the public and economic health and security of America at risk,″ they wrote.
What happened in Michigan?
Michigan is the latest hotly contested state to certify its election result, despite disputes from Mr Trump's campaign.
Mr Biden won in Michigan by more than 154,000 votes. However, Mr Trump claimed voter fraud swayed the outcome, without offering evidence.
The president's Republican allies had called for the certification - which finalises the outcome of a state's popular vote - to be delayed for two weeks to audit ballots in a heavily Democratic county.
However, the Michigan Board of State Canvassers voted to certify the result on Monday. One of two Republicans on the Michigan State Board of Canvassers joined the two Democrats to finalise the outcome.
The other Republican board member, Norman Shinkle, abstained, suggesting delaying the certification over irregularities affecting a few hundred votes in one county.
Mr Biden will now receive all 16 of the state's votes in the US Electoral College, which is expected to approve Mr Biden's victory on 14 December.
Mr Trump's legal team said they would still challenge Michigan's results.
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.
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.
In 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.