The then 25-year-old became queen regent of seven Commonwealth countries including Australia when her father, George VI, died on February 6, 1952.
The governor-general says Queen Elizabeth has a "deep, genuine and warm connection" to the country and its people.
"She is a symbol of our history, represents our modern links to the commonwealth and is an exemplar of service and dedication - characteristics that are important to our national identity."
The governor-general has written to the queen conveying his congratulations "and Australia's ongoing affection and loyalty".
Mr Morrison says the queen's length of service has been extraordinary.
"I have had the great privilege to meet Her Majesty on several occasions and I have never met anyone more impressive, more remarkable," he told reporters on Sunday.
"Her wisdom, her kindness, her sense of duty, is something that I think all of us here in Australia can be very grateful for.
"And so, to you, Her Majesty, from Australia - thank you for all your many years of service and may God save the queen."
The prime minister said it was understandable that Elizabeth did not traditionally mark February 6 with joy, it also being the anniversary of the passing of her father.
"Still, today marks an extraordinary milestone," he said.
"For over half of our history as a federation, the queen has been our monarch. Most Australians have known no other sovereign. She has been a constant presence in a changing world."
Mr Morrison said appropriate Australian celebrations to mark the platinum jubilee would be staged later this year.
The queen's reign has spanned 15 Australian prime ministers, Sir Robert Menzies being the first.
Iconic buildings and monuments in many parts of Australia will be lit up in royal purple on Sunday evening.
Meanwhile, Elizabeth has thanked the public for their support and urged them to get behind the Duchess of Cornwall as Queen Camilla once Charles is crowned king.
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