The commander and his guards were killed in clashes at a security checkpoint, Burhani said, adding that three more servicemen were injured.
"The fighting started at 3.00 p.m. [10:30 GMT] and lasted for two hours. Three people, including a local police commander Mullah Saaduddin, were killed and three others wounded," the spokesman said.
Saaduddin was among the most prominent figures of the Afghan Local Police (ALP).
The Taliban have not commented on the incident so far.