Labor has claimed victory in two South Australian by-elections with rising stars in the party set to replace former Premier Jay Weatherill and his deputy John Rau in state parliament.

With two-thirds of the vote counted in Cheltenham, Mr Weatherill's former seat, SA Unions boss Joe Szakacs had almost 60 per cent of the primary vote and more than 75 per cent with the distribution of preferences.

The result was a near-10 per cent swing to Labor, though a result hardly surprising after the Liberal Party decided not to run endorsed candidates in either seat.

In Enfield, lawyer Andrea Michaels also polled strongly and with 56 per cent counted had almost 48 per cent primary support.

With the distribution of preferences, she was ahead of former Port Adelaide Mayor and independent Gary Johanson 63 per cent to 37 per cent, a near-five per cent swing to the ALP.

Mr Szakacs told supporters he would bring compassion and a "fight for fairness" to his new role.

"In our campaign we talked about great, progressive Labor ideas, about fairness, about the share of justice in our community," he said.

"We will always stand up for working people, their families and our community.

"That is why we won tonight."

Ms Michael's said the results in both Cheltenham and Enfield were a strong indication that people still believed in Labor values.

"They still believe in fairness and opportunity and we've got three years to work on proving ourselves and getting back into government," she said.

Mr Weatherill and Mr Rau announced their retirement from politics late last year, just months after being returned at the 2018 state election.

The former premier said it was somewhat bizarre not seeing his own name on the ticket when he cast his vote earlier in the day.

But even though the Liberals elected not to run, he said Labor had not taken victory for granted.

"By-elections are special cases. Especially when you don't have the Liberal Party contesting.

"It doesn't have the normal argy-bargy that alerts people to the fact that there's an election on."

With Labor retaining both Cheltenham and Enfield, the make-up of the SA parliament will be unchanged with the Liberal government holding 25 seats, the ALP 19 and three in the hands of independents.