The West, particularly France, considers Burkina Faso a key ally in the fight against al Qaeda in the region.
People on Saturday flocked around the headquarters and the French embassy to view the damage.
Burkina Faso's northern border region near Mali is the home of Ibrahim Malam Dicko, an extremist preacher who has claimed responsibility for recent deadly attacks on troops and civilians. The eight attackers of the combined Jihadist attacks were shot dead.
The incident in the early hours came two days after eight soldiers were killed in deadly twin attacks on the French embassy and the Burkinabe military headquarters in the capital Ouagadougou.
Military personnel were on guard outside the headquarters of the country's defence forces in Ouagadougou following the twin attacks on the French embassy and the country's military.
According to French military sources, some 20 jihadists were "killed or captured" on that occasion.
French foreign minister Jean-Yves Le Drian says the attacks targeted several places in Ouagadougou Friday including the French embassy.
"I hope it stops." said Bouri Sawadogo, a local student, "because with the frequency that this is happening we're all scared".
Panicked residents fled the city center on foot or motorbikes as dozens of Burkina Faso soldiers and armored vehicles took up positions in the area.More news: Transgender wrestler Mack Beggs survives first day of state finals
Plumes of black smoke could be seen above the army offices in western Ouagadougou where police and gendarmerie responded. In aiding the fight against terrorist groups, Mr. Kaolaga said, Burkina Faso has become one of their targets. "All day it was like this", said Sylvain Some, who works at a stand near the embassy.
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 Reuters reporter said gunfire had largely ceased by noon.
The capial of Ouagadougou has been attacked by Islamic extremists targeting foreigners at least twice in the past few years. The landlocked country of around 19 million exports cotton and gold.
French President Emmanuel Macron spoke with President Roch Marc Christian Kabore to express his condolences and support, and also to thank the country's forces for their quick intervention.
AQIM merged with other local jihadist groups previous year to form JNIM.
That attack was claimed by the Al-Qaeda-linked Al-Murabitoun group, which was led by one-eyed Algerian jihadist Mokhtar Belmokhtar.
Five countries, Burkina Faso, Niger, Chad, Mali and Mauritania, launched a new task force in 2017 to tackle Islamist militants in the arid Sahel region, to which global donors have committed half a billion dollars.