General Abdul Rahim Mohammed Hussein resigned from his post on Thursday, but the president was not in the country to accept his resignation letter.
The newly built four-storey building, which was designed to house police forensic laboratories, caved in in February as technicians were putting the final touches on the structure.
Nobody died, but at least 12 people, including an Egyptian technician, were trapped under the rubble for several hours.
The minister immediately asked the justice ministry to investigate.
Hussein said he decided to resign after reviewing the results of the probe, which concluded that "there were some shortcomings in the performance of the ministry's engineering organs."
According to Sudanese press reports, the minister also had financial interests in the construction.
Hussein also served as President Beshir's special representative to war-stricken Darfur.