Addressing a crowd in Wilmington, Delaware, Mr Biden pledged to be a president “who seeks not to divide, but to unify” as he dubbed his election win a victory for “the people”.
He said: “The people of this nation have spoken. They’ve delivered us a clear victory, a convincing victory, a victory for you the people.
“I’m humbled by the trust and confidence you’ve placed in me.
“I pledge to be a president who seeks not to divide, but to unify. Who doesn’t see red and blue states, only sees the United States. And who will work with all my heart to win the confidence of all of you, and for that, I believe, is what America is all about.”
Former Tory international development secretary Rory Stewart was full of praise for Mr Biden’s speech, calling it “profoundly needed”.
He said on Twitter: “This is a beautiful speech – fluent, natural and focused on healing not division – deeply reassuring and profoundly needed. Thank you @Joe Biden”
His comments were echoed by Labour MP Dame Margaret Hodge, who said the speech was “Just what we need. Plenty of optimism, decency and hope.”
Former Australian prime minister Kevin Rudd followed a similar tone as he shared a video of fireworks going off in Brisbane, the capital city of his home state Queensland.
He said: “Celebrating last night in Brissie Joe Biden’s win. Or at least borrowing someone else’s fireworks for that purpose A sigh of relief around the world today. A great and healing speech just now from the President-elect. God speed this new administration, both at home and abroad.”
While former Liberal Democrat leader Tim Farron dubbed it a “proper good speech”.
He said: “That was a proper good speech by President Elect Joe Biden, passionate delivery, inclusive, forgiving and welcoming of the other side… even to Trump ‘hey, I’ve lost a couple of elections..’ What a joy to see a grown up back in charge. Phew.”
NBA legend Magic Johnson also hailed Mr Biden’s speech in a statement on Twitter, calling it “everything the American people needed to hear at this time”.
Vice President-elect Kamala Harris’s speech also garnered plenty of attention, with 18-time tennis grand slam winner Martina Navratilova saying people “cannot overstate” the importance of Ms Harris’s election.
She said: “Given that black women are on the bottom of the economic and social rung in this country, Kamala Harris becoming the second most powerful person here in the US and perhaps the world- as a woman of color – this is huuuuuge!!! We cannot overstate this moment !!!!!”
Actresses Halle Berry and Jameela Jamil were also among those to pay tribute to Ms Harris.
Berry said she was inspired by Ms Harris’s “all white suit” while Jamil said Ms Harris and Mr Biden “did a great job”.
She added: “After four years of what felt like a wildfire, it feels like rain has finally come to save us. I have hope again. Let’s hope they come through on their promises. I think they actually care what people think, which makes for a nice change.”