The court action would help stabilize services at the affected hospitals, the department said on Monday.
"This is a victory for innocent and vulnerable patients who suffered the consequences of the strike," spokesperson Foster Mohale said.
But the union said in a statement on Sunday that "no amount of litigation or intimidation by the government will deter us from this noble worthy cause that we have embarked on."
National Education, Health and Allied Workers' Union (NEHAWU) members went on strike last week after wage talks with the government failed.
The health department said on Sunday that clinical operations were hamstrung by a low nursing and administrative support staff turnout.
To mitigate the effects of the healthcare walkout, the South African military had deployed medics to help in the affected hospitals at the request of the health department.
