There are plans in motion for a formal commemoration in Australia.
Mr Morrison said the security of the region was “personal” for Mr Biden, who had two uncles, one of whom was killed, who served in Papua New Guinea during World War Two.
“The relationship goes deep and it is personal, and I think that is something that will bring a lot to an understanding of the people-to-people relationship, and the depth and history of the relationship between our two countries,” Mr Morrison said.
“Anzus has been the bedrock of our security foundations in Australia since that alliance was first established … I look forward to inviting the President-elect to join us next year in their formal capacities at that time and for us to be able to celebrate 70 years of peace and stability and security that has been established by this incredible relationship.”
Mr Morrison did not wait for Mr Trump to concede before congratulating Mr Biden, but paid tribute to his counterpart for showing a “great commitment” to US relationship with Australia.
He thanked Mr Trump, Vice-President Mike Pence and Secretary of State Mike Pompeo for the “very good relationship” between the two administrations during the past four years.
“This relationship is bigger than anyone individual, and those who have the great privilege to serve in either the offices of the prime minister or president are the custodians of that enduring relationship,” Mr Morrison said.
“The United States is one of the world’s greatest democracies, alongside Australia and many others, and democracy has proven – not just in the times of still waters but when the waters can get choppy – the process they have always stood by to resolve such differences.”
Mr Trump has declared he will fight the election result in many states through the courts and had not conceded when Mr Biden claimed victory.
But Mr Morrison said he had “great confidence” in the American democracy and said it was important to express patience and respect for the US system.
“There is much work for like-minded countries like Australia and the United States to get on with and those processes, and the United States will come to their conclusion and the transition will proceed as we understand it to be,” he said.
“This is not a new process; it is a time honoured and time established process and I have confidence it will resolve itself in time.”
Federal opposition leader Anthony Albanese said he had met Mr Biden and said he was “a friend of Australia”.
“The United States alliance is our most important one,” Mr Albanese said. “Partners on security issues, partners on the economy and the trade relationship.”
Rob Harris is the National Affairs Editor for The Sydney Morning Herald and The Age, based at Parliament House in Canberra