The coordinated operation underscored the fragility of the Sahel nation, one of a string of African states struggling with a bloody jihadist insurgency.
Heavy gunfire broke out mid-morning in the centre of the capital Ouagadougou, an AFP reporter heard.
Witnesses said five armed men got out of a car and opened fire on passersby before heading towards the French embassy. The car was later seen ablaze.
At the same time, an explosion occurred near the headquarters of the Burkinabe armed forces and the French cultural centre, which are located about a kilometre (half a mile) from the site of the first attack, other witnesses said.
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"Four attackers were neutralised in the attack on the French embassy," it said in a statement posted on its Information Service website.
"Special units of the defence and security forces have been deployed," it said, adding that a complete toll was not immediately available.
In Paris, a French government source described the situation as "under control." President Emmanuel Macron's office said the French leader was "being informed in real time" of the situation.
Burkina Faso is one of a group of fragile countries on the southern rim of the Sahara that are battling jihadist groups.
On August 13 last year, two assailants opened fire on a restaurant on Ouagadougou's main avenue, killing 19 people and wounded 21. The attack remains unclaimed.
On January 15 2016, 30 people, including six Canadians and five Europeans, were killed in a jihadist attack on a hotel and restaurant in the city centre.
Responsibility was claimed by a group called Al-Qaeda in the Islamic Maghreb (AQIM).
France, the former colonial power in the Sahel region, has deployed 4,000 troops and is supporting a five-country joint force gathering Burkina Faso, Chad, Mali, Mauritania and Niger.
The United Nations also has a 12,000-strong peacekeeping force in Mali called MINUSMA, which has taken heavy casualties. Four UN peacekeepers were killed by a mine blast on Wednesday in the centre of the country.
In a separate development on Friday, the specialist US website SITE, which monitors jihadist activity, said kidnappers had released a video of a 75-year-old French hostage, Sophie Petronin, who had been abducted in northern Mali in late 2016.
Her kidnapping, hitherto unclaimed, was carried out by the "Support Group for Islam and Muslims." In the background, Macron's voice is heard on a loop, saying "I will protect you.
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