Continued counting should bring into sharper focus the size of the swing involved in the victory.
The Liberals are expected to retain at least 15 seats with independents poised to take the remaining five.
The Electoral Commission on Sunday morning had Labor's primary vote up by eight percentage points to 40.5 per cent with the Liberals on 34.6 per cent.
More counting could give a clearer picture in five seats still considered doubtful, including Dunstan, held by outgoing premier Steven Marshall.
In a victory speech to the Labor faithful at Adelaide Oval on Saturday night, Mr Malinauskas thanked the people of South Australia for their support.
"It is not lost on me the significance of the privilege and the size of the responsibility that you have invested in me and my team," he said.
"It means that we've got a big job to do."
Conceding defeat, Mr Marshall said it had been an honour and privilege to serve as premier.
"It's great to live in a country where we have free and open elections," he told supporters.
"Today the people of South Australia have spoken. They've elected a new government."
Mr Marshall gave no indication whether he would stay on as opposition leader should he retain his seat.
Among the independents returned were Speaker Dan Cregan in the Adelaide Hills, Geoff Brock, who shifted to the seat of Stuart in the mid north, and Troy Bell in Mt Gambier.
Fraser Ellis was also voted back in Narungga, which takes in Yorke Peninsula.
Both Mr Cregan and Mr Ellis were elected as Liberals in 2018 but left the party last year to sit on the crossbench.
In other results, Nick Champion successfully switched from the federal parliament to the state house, coasting to a win in the northern Adelaide seat of Taylor.Â
Labor's victory in SA was the first against an incumbent government at any state or territory election during the COVID-19 pandemic.
