SAA, which has not generated profit since 2011, is regularly cited by ratings agencies as a drain on the government's purse. It made a loss of 1.5 billion rand in 2015/16.
Jarana told a news conference on Thursday that SAA expected to break even by 2021 and that it was in talks with unions about reducing staff costs.
Selling a stake in the airline to an equity partner - a long-held plan which was discussed as far back as 2016 - was about two years away, he added.
This week the head of the Treasury said selling 49 percent of SAA was a theoretical example of how the government could narrow the budget deficit.
Finance Minister Nhlanhla Nene told the same news conference that the turnaround strategy would include a review of the airline's fleet and loss-making routes.