Gigaba told a media briefing on the state of the economy that a successor would be appointed at the upcoming annual general meeting of the national airline.
National Treasury would also move to have the post of CEO of the airline, which has been vacant for nearly two years, filled.
"At the upcoming AGM, we will attend to the matter of appointing her successor. We will be recommending a chief executive to Cabinet in the coming weeks."
Gigaba said though SAA remained in line for a cash injection from National Treasury, it would come with stringent conditions, notably a commitment to finally implement the company's turnaround strategy.
"Whatever assistance we give them is certainly not going to come easy and freely. We are going to be hard with SAA and the turnaround is not going to help them quickly therefore the conditions we are going to set for them are going to be tough. They are going to have to demonstrate that they are prepared themselves to regain the confidence of the public," he said.
"The strengthening of the balance sheet of SAA must be linked fundamentally to the organisation returning to proper governance and demonstrating that it is going to now implement the turnaround strategy rather than abdicate from that."
Gigaba said he would make an announcement on SAA in his medium-term policy statement in October, and confirmed that he was also discussing financial support for the South African Broadcasting Corporation with Communications Minister Ayanda Dlodlo.
Dlodlo this week declined to tell MPs the size of the lifeline the SABC is seeking. Gigaba said a capable board also needed to be appointed, and added pointedly that it must focus not on the politics of the country but on the task of running the company in a viable fashion.
He also called on Public Enterprises Minister Lynne Brown to act with speed to replace the Eskom board, which has been without a chairperson since Monday when Ben Ngubane resigned under a cloud, and to ensure the vacant seat of CEO be filled following Brian Molefe's departure.
"We are very concerned about the state of state-owned companies, particularly their finances and the contingent liabilities," he said.
"The minister of public enterprises should act speedily to strengthen the Eskom board.
"If we address the governance issues at Eskom....if we build confidence as far as the executive directors and non-executive directors are concerned, Eskom will fall off the list of major concerns.
"It will become normal."
He said there would then be scope to scale down the power utility's government guarantees, which stands at R350 billion.
"We are looking forward to that taking place."
He hinted that given the recession, there may not be economic space for the National Energy Regulator of SA (Nersa) to approve Eskom's application for tariff increases of almost 20% for the coming year.
"A decision will have to be taken which does not jeopardise the economy and the position of consumers."