Biden said the US faced a "very dark winter" and the "worst wave yet", and Americans had to put aside political differences to tackle Covid-19.
He has named a new task force and vowed to "follow the science" as he puts together his transition team.
Donald Trump still refuses to concede defeat and is challenging key results. He is taking legal action in several states.
Biden's victory was declared on Saturday but it remains a projection, with a number of states still counting ballot papers. He leads Trump in the nationwide vote by about 4.5 million.
Biden and Vice-President-elect Kamala Harris have launched a website for the transition, saying the team will also focus on climate change, the economy and tackling racism.
But the president-elect will need the help of an agency called the General Services Administration to begin the transition process and its Trump-appointed head has given no indication when that will happen.
What are the Covid plans?
In a TV address, he said "I implore you, wear a mask", calling it the "single most effective thing we can do to stop the spread of Covid".
He added: "A mask is not a political statement but it is a good way to start pulling the country together."
During the election campaign, his mask-wearing was markedly different to Trump, whose attitude to it has varied considerably.
They have differed too on how to follow scientific advice. Biden said on Monday there would be a "bedrock of science" to his policy. President Trump's comments on Covid-19 have often conflicted with scientists, including leading infectious disease expert Dr Anthony Fauci.
Biden said that, as president, he would improve contact-tracing and supplies of protective equipment, and address the racial disparity of coronavirus infections.
"This is a crisis that affects everyone and I will be a president for everyone," he said. "We can get this virus under control, I promise you."
He earlier named three co-chairs and 10 members of his task force. Among the co-chairs named is Vivek Murthy, who was appointed US surgeon-general by President Barack Obama in 2014 and removed by President Trump in 2017.
One member is immunologist Rick Bright, who says he was ignored and then removed by the Trump administration over his early warnings on Covid.
Biden also welcomed the news from Pfizer and BioNTech that preliminary analysis showed their vaccine in development could prevent more than 90 percent of people from getting Covid-19.
But he also warned it was "important to understand that the end of the battle against Covid-19 is still months away".
BBC North America correspondent Anthony Zurcher says that Biden's choice to address the pandemic as the first public event for his new transition team afforded him an opportunity to look and act like a man who will soon be president, despite Donald Trump's decision to not concede defeat.
Covid cases in the US since the epidemic began are nearing 10 million, and there have been more than 237,000 deaths recorded so far, Johns Hopkins research shows.
Donald Trump also welcomed the vaccine statement, tweeting it was "such great news".