Quoting family members, SM Badiuzzaman, officer-in-charge of Sadar Police Station, said the couple had an affair in Dhaka where they work.
They lived at a rented flat together in Shewrapara of the capital.
Both of them returned home after the government announced a general holiday and had not been in touch with each other since.
On Wednesday, the woman went to his lover’s house and started a fast-unto-death unless he agreed to marry her.
But police intervened and brought the couple to the police station.
Later, both families met and went ahead with the marriage ceremony near Akbaria Hotel adjacent to Sadar Police Station on Wednesday night.
The government has shut down transport and cancelled programmes to prevent the transmission of coronavirus. The people have been advised to avoid gatherings and stay indoors.
Bangladesh recorded 1,231 coronavirus cases and 50 deaths until Wednesday morning.