Mkhize did not report any new deaths from the disease, with the death toll standing at 25, as announced on Saturday, Xinhua news agency reported.
The rapid rise in the confirmed cases might be the result of more tests being conducted after the country rolled out mass community testing last week. The total number of tests conducted to date is 80,085, Mkhize said.
"We are also seeing an increase in tests conducted at public laboratories. Of the 5,032 tests conducted in the past day, 3,192 were done in public laboratories," the minister said.
Gauteng remains as the province with the highest number of 895 confirmed cases, followed by Western Cape Province with 587 cases and 443 in KwaZulu-Natal Province.
The epidemic has also spread to a local prison, infecting 23 officials and three inmates.
The cases in the prison were detected after mass testing at the East London Correctional Facility in Eastern Cape Province, which recorded its first case on April 6, according to the Department of Justice and Correctional Services.
With more results to be released on Tuesday, the number of confirmed cases is expected to rise, the department said.
"More testing for both officials and inmates in the entire East London Management Area is continuing and designated quarantine and isolation areas have been identified," said the department.
The deadly contagion has infected at least 1.8 million people across the globe, while as many as 112,241 people have succumbed to it, according to the data compiled by Johns Hopkins University.
( With inputs from ANI )