The prime minister confirmed the conversation on Tuesday in a tweet, with No.10 adding he had also passed on good wishes to vice president-elect Kamala Harris.
Johnson said he looked forward to working with the new US administration on “shared priorities”, which include climate change, as the UK continues preparations to host the COP26 summit in Glasgow next year.
Johnson said: “I just spoke to Joe Biden to congratulate him on his election.
“I look forward to strengthening the partnership between our countries and to working with him on our shared priorities – from tackling climate change, to promoting democracy and building back better from the pandemic.”
A Downing Street spokesperson added: “In a call this afternoon the prime minister warmly congratulated Joe Biden on his election as president of the United States.
“The prime minister also conveyed his congratulations to vice president-elect Kamala Harris on her historic achievement.
“They discussed the close and longstanding relationship between our countries and committed to building on this partnership in the years ahead, in areas such as trade and security – including through NATO.
“The prime minister and president-elect also looked forward to working closely together on their shared priorities, from tackling climate change, to promoting democracy, and building back better from the coronavirus pandemic.
“The prime minister invited the president-elect to attend the COP26 climate change summit that the UK is hosting in Glasgow next year. They also looked forward to seeing each other in person, including when the UK hosts the G7 summit in 2021.”
The call is thought to have taken place at around 4pm and lasted around 25 minutes.
Biden had also spoken with French president Emmanuel Macron, German chancellor Angela Merkel and Canadian prime minister Justin Trudeau on Tuesday.
The conversation follows an embarrassing episode in which the government blamed a “technical error” for the word “Trump” appearing faintly in the background of Johnson’s first tweet congratulating Biden and Harris.
It appeared as though a previous message had been overwritten.
It comes amid fears over the future of the so-called special relationship between the US and UK, with the Democrats not thought to be supportive of Brexit.