"She entered Turkey on 2 January from Madrid. There are images of her at the airport," Cavusoglu was quoted as saying by state-run news agency Anatolia.
Telephone records show she crossed into Syria six days later, he said.
Cavusoglu said the 26-year-old, who had married Coulibaly in an Islamic ceremony, stayed at a hotel in the Kadikoy district of Istanbul and was accompanied by another person.
He did not give further details on the identity of the other individual and did not make clear if she had travelled to Syria on her own.
Cavusoglu added that Turkey passed the information to the French authorities "even before they asked for it" as soon as Ankara identified her whereabouts.
"We told them: 'The person you are looking for was here, stayed here and crossed into Syria illegally'," he said.
Interior Minister Efkan Ala also said Turkey did not refuse Boumeddiene entry because the French authorities had made no such request and that they had not warned Ankara that she was dangerous.
He added that Turkey's intelligence agency, the National Intelligence Organisation (MIT), and the police were still working to shed more light on the matter.
A Turkish security source earlier told the AFP news agency that Boumeddiene had entered Turkey on 2 January and was believed to have moved on to the south-eastern Turkish city of Sanliurfa and then to Syria.
Turkey did not arrest her because of a lack of timely intelligence from France, the source said.