While Mr Biden avoided referring to President Donald Trump directly, he said that as president he hopes to restore decency, defend democracy "and give everybody in this country a fair shot."
Mr Biden said he will now turn his attention to the battle against COVID-19, securing healthcare, getting climate change under control and achieving racial justice.
The president-elect also thanked his running mate Kamala Harris and recognised the historic nature of her election as vice president.
"Don't tell me it is not possible in the United States."
Mr Biden also acknowledged keen international observers who have been watching the election from abroad.
"Tonight, the whole world is watching America and I believe that at our best, America is a beacon for the globe. We will not leave, we will lead by the example not only of our power but by the power of our example," he said.
After jogging onto the outdoor stage to the sounds of Bruce Springsteen in his hometown of Wilmington, Delaware, Mr Biden delivered a message of hope and healing to a crowd of cheering supporters and tens of millions more on television.
The victory speech followed a desperately bitter election conducted in the midst of a raging coronavirus pandemic.
But instead of sounding triumphant, Mr Biden's accent was more on changing hearts in a country split down the middle between Democrats and Republicans.
"Let this grim era of demonization in America begin to end, here and now."
Mr Biden, who is 77 and turns 78 later this month, will be the oldest person to become president when he takes office on 20 January.
Earlier, Kamala Harris opened the speech by reflecting on her own groundbreaking win.
Joe Biden declares 'now is a time to heal' in acceptance speech for US President-elect
"Every little girl watching tonight sees that this is a country of possibilities. And to the children of our country, regardless of your gender, our country has sent you a clear message. Dream with ambition. Lead with conviction," she said.
Ms Harris dedicated the victory to her deceased mother, who came to the US from India at the age of 19.
"She believed so deeply in an America where a moment like this is possible. I am thinking about her and about the generations of women.. who.. have paved the way for this moment tonight," she said.
"While I may be the first woman in this office, I will not be the last."
Joe Biden will become the 46th president of the United States and is due to be sworn in with his running mate vice president-elect Kamala Harris on 20 January.
Additional reporting by AFP.