A post-mortem examination found she died from injuries "consistent with compression of the neck".
The suspect was detained in Tallinn just before 12.30pm on Wednesday and is being held by police there ahead of extradition proceedings back to the UK.
Scotland Yard said it had worked with Eurojust in the Hague and Europol to trace and detain the 32-year-old man after issuing an European arrest warrant.
Ms Garcia-Bertaux, a French national who had been living in the United Kingdom for many years, did not turn up for work at public relations firm Golin on 4 March and was reported missing the following day.
A murder investigation led by Detectives from the Met's Homicide and Major Crime Command is underway.
The 34-year-old French national's remains were discovered on 6 March in the garden of her home address in Darell Road, where she had lived since April last year.
Police arrest suspect, 32, in Estonia over murder of French film producer
Garcia-Bertaux had studied performance design practice at Central Saint Martins in London before working as an executive assistant at Golin.
As well as being an accomplished dancer, she was also a talented singer-songwriter and played in a cover band called The Tams.
Last year, she assisted on short film Gerry, starring Joan Collins, who has since paid tribute to Laureline. Laureline was passionate about film.
The film legend, 85, said: "I'm shocked by the horrifying news".
Jonathan Hughes, chief executive of public relations company Golin where she had worked since January, said: 'On behalf of everyone at Golin I want to express our deepest sympathies to Laureline's family and friends. "Laureline was a wonderfully creative, caring and charismatic woman, full of energy and joie de vivre", he said.