It also said detectives arrested two suspects over the double murder.
Police recovered the bodies of Saheda Begum alias Rahima Begum, 65, and Sumi, 20, from a flat of a four-storey building in the city's Mirpur-2 area on Tuesday evening.
A team of DB (West Division) arrested Md Yusuf Khan, 20, and a boy, aged, 14, from Sadarghat area on Wednesday afternoon and recovered the mobile phone of one of the victims, Tk 14,000, three imitation chains and a pair of earrings from their possession.
Briefing reporters at the DMP Media Centre on Thursday, Additional Commissioner (DB) Abdul Baten said they started investigation into the double murder soon after the incident.
During investigation, they came to know that the arrested duo went to the rented flat of Rahima on Monday night for antisocial activities, he said.
Yusuf slept with Sumi in a room while the minor boy in a balcony of the flat, Baten said, adding that they had an altercation over the payment.
The duo strangulated Sumi to death on Tuesday morning fearing that they might be beaten over the payment, he said.
The DB official said one of the arrestees then called Rahima out of her room and they also strangled her. They took away the money and imitation ornaments and one mobile phone set while fleeing the scene.
Rashida Begum, daughter of victim Rahima, filed a murder case with Mirpur Police Station on Tuesday night. Later, the case was transferred to the DB police for investigation.