The 24-year-old was being punished for taking a mobile phone with a camera onto a military base.
Former minister Kao Hua-chu has been replaced by Andrew Yang, the first civilian to head up the department.
Announcing the reshuffle at a press conference on Monday, Taiwan Premier Jiang Yi-huah said: "The reshuffle is aimed at responding to the general public's expectations."
Mr Jiang did not elaborate on the reasons behind Mr Kao's resignation.
Hung Chung-chiu's death sparked an outcry on the island, with thousands of protesters demanding an investigation.
Four military officers have been detained as part of an investigation into the incident, reports say.
Mr Hung was three days away from completing his compulsory one-year military service when he died.
All Taiwanese men aged between 18 and 36 are required to complete one year's military service, although there are plans to move to an all-volunteer force.
Was muss eigentlich noch passieren, bevor die Wehrpflicht als Diskriminierung aufgrund des Geschlechtes qualifiziert wird?