The coordinated attacks in Ouagadougou, which coincided with a meeting of regional anti-jihadist forces, underlined the struggle the fragile West African nation faces in containing a bloody and growing jihadist insurgency.
Eight armed forces personnel were killed, the government said, while a French security source said 12 more were seriously injured. Earlier security sources had reported a higher toll.
There was no immediate claim of responsibility, and while a jihadist plot is the most likely, the government was not ruling out the involvement of plotters behind a failed coup in 2015.
Two people were arrested near the headquarters, a security source told AFP.
The government said the attack on the military HQ was a suicide car bombing and that the G5 Sahel regional anti-terrorism force may have been the target.
"The vehicle was packed with explosives" and caused "huge damage", Security Minister Clement Sawadogo said.
Prime minister Paul Kaba Thieba described "truly apocalyptic scenes" in the aftermath of the attack.
"Obviously that will not happen."
Officials from Burkina Faso, Chad, Mali, Mauritania and Niger had gathered for a G5 Sahel meeting, an anti-terrorist group designed to combat jihadists on the southern rim of the Sahara.
The location of the meeting was changed at the last moment, Sawadogo said, adding the room they had been due to use at the military HQ was blown up.
"Perhaps it was the target. We do not know at the moment. In any case the room was literally destroyed by the explosion," the minister added.
Witnesses said five men armed with Kalashnikov assault rifles got out of a car and opened fire on passersby before heading towards the French embassy.
They were "dressed in civilian clothes" with their faces uncovered, witnesses said.
At the same time, the bomb went off near the headquarters of the Burkinabe armed forces and the French cultural centre, about a kilometre (half a mile) from the site of the first attack.
A gunman who attacked the military HQ was wearing the uniform of the national army, according to a security source.
"After soft targets, such as hotels and restaurants, this attack aimed at hard targets, strong symbols," Paul Koalaga, a security consultant in Burkina Faso said, adding there appeared to be "a problem at the intelligence level".
The G5 Sahel force aims to train 5,000 troops and to be fully operational by the end of the month.
It has already carried out operations against jihadist fighters with help from the French army.
Mahamadou Issoufou, Niger's president and the current chair of the group, said Friday's attacks "will only strengthen the resolve of the G5-Sahel and its allies in the fight against terrorism".
The insurgency in the region has caused thousands of deaths, prompted tens of thousands to flee their homes and dealt crippling blows to economies that are already among the poorest in the world.
Burkina Faso has been the target of jihadist attacks since 2015, and this is the third time in two years that Ouagadougou is the target of jihadist attacks targeting places frequented by Westerners.
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.
That attack was claimed by the Al-Qaeda-linked Al-Murabitoun group, which was led by the one-eyed Algerian jihadist Mokhtar Belmokhtar.
A group called Al-Qaeda in the Islamic Maghreb (AQIM) also said some of its militants were involved.
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