Local security sources put a still-incomplete death toll at about 15 while other sources reached by AFP from Paris sketched an even bloodier outcome, with at least 28 people killed in the second assault alone.
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 Burkinabe capital Ouagadougou.
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.
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Burkinabe security sources said around 15 people were killed, while the army's medical chief, Colonel Amado Kafando, said 75 others had been injured.
It was unclear whether these figures applied to one or to both attacks, and whether they included the assailants.
Earlier, the government had announced that six assailants had been killed -- four in the embassy attack and two in the military attack -- while state TV said seven members of the security forces had also died.
French government sources said there had been no French casualties and described the situation in Ouagadougou as "under control".
Information Minister Remis Fulgance Dandjinou said the attack "has strong overtones of terrorism." The Paris public prosecutor said it had opened a formal investigation into "attempted murder in relation to a terrorist enterprise", an expected procedural step after attacks targeting French citizens or interests.
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".