French President Francois Hollande thanked Nigeria's authorities for helping secure the release of Francis Collomp, 63, in the city of Zaria.
Unofficial reports say he managed to escape from his cell during an army operation against the militants.
The French foreign minister is being sent to Nigeria to meet him.
The news comes after four other French hostages were released in neighbouring Niger having been held for three years by gunmen with links to al-Qaeda.
There were reports that at least 20m euros (Â£16.7m; $27m) were paid in ransom for the four. The French government denied any public money was used.
Since then a French Roman Catholic priest has been abducted in Cameroon, another country in the region.
Mr Collomp was kidnapped on 19 December last year by armed men who attacked the residence of his employer, the French wind turbine manufacturer Vergnet, in the north Nigerian state of Katsina.
Ansaru, a militant group linked to the Islamist Boko Haram movement, said it had carried out the abduction.
Mr Hollande said in a statement on Sunday:  "The president greets with joy the release of our compatriot Francis Collomp.
"France had never ceased to make every effort to achieve this happy outcome. The president expresses all of his gratitude to Nigerian authorities, with whom France worked in close cooperation, for their decisive action."
He said he was sending Foreign Minister Laurent Fabius to Nigeria to greet the former captive.
A source "close to the case" told AFP news agency that Mr Collomp had fled during an exchange of fire between the army and militants after his cell door was left open.
Reuters news agency quoted an unnamed French government source who also said the captive had escaped. However, it added that a source in the French foreign ministry had denied the report.
A diplomatic source told Reuters that Mr Collomp was weak and had lost a lot of weight but was uninjured.
Seven French citizens, Mr Hollande said, remained in captivity in Syria, Mali and Nigeria.
It is believed that Fr Georges Vandenbeusch, who was abducted in northern Cameroon on Thursday, has been taken into Nigeria.
According to AFP, the priest was abducted by Boko Haram, with the help of Ansaru.
A French hostage held by Islamist militants in northern Nigeria for nearly a year is free after reportedly escaping during a shoot-out.