He said Durban metro police officers were deployed throughout the city.
"There are people who are complying and who have permits. These are those who need to go to work as part of essential services. But there were also people out on the streets when they shouldn’t be," Zungu said.
He said it was too early to confirm arrest numbers.
Police spokesperson in KwaZulu-Natal Col Thembeka Mbele said officers were out working together to "ensure the lockdown is strictly adhered to".
Videos of some offenders being dealt with by police were circulating on social media.