"The situation is under control at French diplomatic premises," the source said after the attack which also targeted the country's military headquarters.
Burkina Faso's government said four attackers were killed in the assault on the embassy.
French Foreign Minister Jean-Yves Le Drian told LCI television that there were "no French casualties, as far as we're aware" and that the attackers had been "neutralised", a military euphemism for killed.
President Emmanuel Macron's office said he was being kept informed in real time of the situation.
"French nationals in Ouagadougou should follow the embassy's instructions," the presidency said.
The embassy told them to remain indoors.