Labor swept to power in the state on Saturday under the leadership of Peter Malinauskas after only one term in opposition.
Outgoing premier Mr Marshall's loss was the first for any incumbent state or territory government during the pandemic.
Prime Minister Scott Morrison's own leadership will be put to the test within months with a federal election due in May.
"Our election was always going to be a challenging one and I don't think we have ever underestimated the challenges that lie ahead," South Australian Senator Birmingham told ABC's Insiders program on Sunday.
"But we also remain very determined to make sure Australians understand the choice ahead of them."
He said in terms of the South Australian result, a number of factors were at play.
"We saw the Labor party run a very targeted, very singularly focused campaign around hospitals and ambulances," he told the 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, it was 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.
