The prime minister's own leadership will be put to the test within weeks with a federal election due in May.
Labor swept to power in the state on Saturday under the leadership of Mr Malinauskas after only one term in opposition.
Outgoing premier Steven Marshall's loss was the first for any incumbent state or territory government during the pandemic.
Mr Morrison congratulated Mr Malinauskas and his wife by phone.
"It's a significant victory and they are elected with a very strong mandate to move forward with many issues that they've intended to take forward," he told reporters in Sydney on Sunday.
But the prime minister was quick to turn the focus on his own upcoming battle.
"What I know is is that Anthony Albanese is not Peter Malinauskas," Mr Morrison said.
"There is a big difference between Anthony Albanese as the federal Labor leader and what we see in the performance of some of his state colleagues."
But Wayne Swan, the former Labor treasurer and now ALP national president, said the SA result sends a clear message that voters have had enough of Liberal governments that are out of touch with their lives.
"Whilst the failures and mistakes of Premier Marshall played a big part, the person who has done the most damage to the Liberal Party brand in Australia in the last three years is Scott Morrison," Mr Swan said in a statement.
"This result should have all of his MPs trembling."
Federal finance minister and SA senator Simon Birmingham blamed a Labor scare campaign and the unlucky timing of the Omicron COVID-19 variant for Mr Marshall's election loss.
"We saw the Labor party run a very targeted, very singularly focused campaign around hospitals and ambulances," he told ABC's Insiders program.
"I think there were many misleading aspects to that campaign, the electoral commission found so in the last day or so."
He also believes COVID-19 played a difficult role for Mr Marshall because when he opened SA's borders to the rest of the country on November 23, only a day later the World Health Organisation reported the Omicron variant as a concern.
"You couldn't have had perhaps a more unlucky timing than Steven Marshall faced in that regard," Senator Birmingham said.
"The carefully calibrated plans he had for reopening were clearly blown out of the water at that time."
Otherwise, he believes history will judge Mr Marshall's government and its policy achievements more kindly than the electorate did.
Don Farrell, an SA Labor senator and mentor to Mr Malinauskas, said he was "overwhelmed and proud" that he is now premier.
"One of the good things about him, is that when he says he is going to do something, he does it," Senator Farrell told Sky News' Sunday Agenda program.
The business sector was also quick to congratulate the new premier.
"It's been a difficult few years and now that South Australians have made their decision, we can all focus on creating the best business environment possible in the face of challenges such as skills shortages, supply chain and climate change issues, geopolitical upheaval and rises in inflation," Ai Group state head Jodie van Deventer said in a statement.
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