They had fled across the border to Bangladesh amid heavy fighting between the Myanmar military and rebel insurgents in Rakhine
The minister gave his sample to the Institute of Epidemiology, Disease Control and Research or IEDCR for a COVID-19 test on Tuesday after showing symptoms of the disease.
His test result came back positive on Wednesday, his ministry said in a media statement.
Following instructions from Prime Minister Sheikh Hasina, Shahab Uddin will be admitted to the Combined Military Hospital (CMH) in Dhaka.
The 65-year-old minister had sought blessings from the people for a quick recovery from the disease.