The suspect, Cheikh Tidiane Diaw, was arrested at his home Wednesday night, chief prosecutor Giuseppe Creazzo announced at a press conference.
DNA evidence linked to the suspect was found in a condom and on a cigarette at Olsen's house, where Diaw's fingerprints were also found, Creazzo said. Diaw had taken Olsen's phone and put his SIM card in it, and was using the device when he was arrested, the prosecutor said.
Police believe that Olsen and Diaw, an illegal immigrant from Senegal who was born in 1988, had met at the city's Montecarla nightclub and left to go back to Olsen's place together, where they had consensual sex, Creazzo said.
He said Olsen was strangled by something "other than bare hands," but also had two fractures to her cranium.
Police have established there was no one else present in the room when Olsen died, Creazzo said. Authorities are awaiting the results of toxicology tests.
Olsen was found dead in her apartment on Saturday, after her boyfriend had been unable to reach her and asked Olsen's landlord, Claudia Colivicchi, who lived upstairs, to open the apartment for him.
Colivicchi told Italian media outlets that when she opened the door, Olsen was on a bed in a mezzanine loft of the studio apartment.
Olsen's dog, Scout, who was a regular feature on her Instagram and Facebook feeds, had defecated in multiple places within the apartment, indicating that the well-trained beagle had not been taken for a walk in some time, Colivicchi told media outlets.
The woman's friends told investigators they were with Olsen at the Montecarla nightclub in Florence on Thursday night and left her there between 3 a.m. and 4 a.m. Friday, local media reported.
Olsen was originally from Summer Haven, Florida. She was living in Florence, where her father, an architect, works at an art school. Her father was seen with friends outside her building Monday, leaving flowers.
"The police haven't really said anything yet. My son is waiting for the police to share more," Olsen's grandmother, Ann Olsen, told CNN by phone from Florida.
Olsen's friends told CNN affiliate WJAX in Jacksonville, Florida, that Olsen was "full of life."
"She never met a person who didn't like her. ... She had the best personality," Staci Kelly told the station, adding that Olsen loved to travel. "She never really stayed in one spot for too long."