The former UK foreign secretary was elected as party leader by Conservative Party members following the resignation of Boris Johnson.
Mr Albanese said he is set to speak to the new prime minister within the next day.
"I congratulate Liz Truss on her election as leader of the Conservative Party," Mr Albanese told parliament on Tuesday.
"It is a special privilege to be chosen to lead your party and it is an extraordinary honour to have the chance to lead your country."
Ms Truss will be the fourth UK prime minister in the past six years, and just the third woman in the role.
Mr Albanese said he had previously met Ms Truss during the recent NATO summit in Spain.
"Ms Truss will walk through that famous black door of number 10 Downing Street at a time when the people of Britain are facing many serious challenges," he said.
"Out of the deep and abiding affection that Australians have for our oldest ally and oldest friends, we wish her well in the service of her country."
Opposition Leader Peter Dutton has also paid tribute to the new British prime minister.
"Liz is an incredibly intelligent, pragmatic and courageously leader. She will make a fine prime minister," he told parliament.
"In these testing times, Liz will be the strongest of leaders to advance our two nations' mutual security and economic objectives including most importantly through AUKUS."
Mr Dutton also paid tribute to Mr Johnson, whom he called a true friend of Australia.
Opposition foreign affairs spokesman Simon Birmingham said Australia would have a lot of confidence in the new UK prime minister.
"We will have a very strong working relationship between our two nations and working with other nations of the world," he told ABC TV.
"On the world stage we have seen that Liz takes a very strong, principled stance in defence of multilateral institutions and the rules-based order against autocracy and others who have sought to disrupt those."
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