Biden spent much of his election campaign criticising President Donald Trump's handling of the pandemic, which has now caused the deaths of 237,000 people in America.
The United States saw a record number of new infections last week, with the total number of cases nearing 10 million.
After four days of uncertainty as votes were counted in key states, Biden's victory in Pennsylvania put him over the threshold of 270 Electoral College votes he needed to clinch the presidency. Trump has not conceded and has vowed to challenge the outcome in court.
The coronavirus task force will be charged with developing a blueprint for containing the disease once Biden takes office in January.
It will be headed by three co-chairs, former surgeon general Vivek Murthy, former Food and Drug Administration commissioner David Kessler, and Dr. Marcella Nunez-Smith of Yale University, according to two people familiar with the matter.
"I will spare no effort - or commitment - to turn this pandemic around," Biden said in his victory speech on Saturday in Wilmington.
In his speech Biden pledged that as president he would seek to unify the United States and "marshal the forces of decency" to battle the coronavirus pandemic, restore economic prosperity, secure healthcare for American families and root out systemic racism.
Biden made an explicit call for cooperation between America's two major political parties as he faces political dysfunction and partisan gridlock in Washington.
"Let this grim era of demonisation in America begin to end here and now. The refusal of Democrats and Republicans to cooperate with one another, it's not some mysterious force beyond our control. It's a decision, a choice we make. And if we can decide not to cooperate, then we can decide to cooperate."
Trump's allies made it clear the president does not plan to concede anytime soon. One Trump loyalist said the president was not ready to admit defeat even though there would not be enough ballots thrown out in a recount to change the outcome. "There's a mathematical certainty that he's going to lose," the loyalist said.
Trump has filed a raft of lawsuits to challenge the results, but elections officials in states across the country have said there has been no evidence of significant fraud, and legal experts say Trump's efforts are unlikely to succeed.
Symone Sanders, a senior Biden adviser, told CNN's "State of the Union" program on Sunday that "a number of Republicans" have reached out to the president-elect but "I don't believe anyone from the White House has."
Sanders sidestepped a question on whether Biden planned to sign a series of executive orders shortly after taking office on 20 January that would reverse several contentious Trump policies.
The Washington Post reported on Sunday that Biden plans to sign orders repealing a ban on travellers from several Muslim-majority nations, rejoining an international climate accord, reversing Trump's withdrawal from the World Health Organization and buttressing a program protecting from deportation "Dreamers" immigrants brought to the United States illegally as children.
The Covid-19 task force announcement will kick off a busy week that will see Biden and Vice President-elect, Kamala Harris, moving forward with the presidential transition on a number of fronts, with Biden turning in earnest to the task of building his administration ahead of his January inauguration.
His transition team has launched a new website, BuildBackBetter.com, and a new social media handle, @transition46, to provide the public with information on the handover.