The group, led by the Malian jihadist Iyad Ag Ghaly, said it carried out the attacks on the French embassy and Burkina Faso's military HQ in the capital Ouagadougou on Friday, which left eight soldiers dead.
JSIM said it took the action in response to the deaths of some of its leaders "in a French army raid in northern Mali two weeks ago," the private news agency reported.
According to French military sources, some 20 jihadists were "killed or captured" on that occasion.
JSIM had already claimed responsibility for a February 21 attack near the border with Niger which left two French soldiers dead and a third injured in an area which is believes to shelter jihadists.
Friday's attacks in Ouagadougou on the French embassy and the country's military HQ left 12 people seriously injured according to an official toll while eight assailants were killed, along with the soldiers.
The coordinated attacks coincided with a meeting of the G5 Sahel regional anti-jihadist force under French command