An attack on the military headquarters strikes at the heart of Burkina Faso's security establishment, while the French embassy is normally one of the safest places in the capital.
Friday's attack was the second such incident in Burkina Faso since gunmen attacked the Splendid Hotel in the heart of the capital in 2016, killing at least 30 people and wounding over 50 more.
Four of the suspects were killed at the embassy while at least two others were also killed at the military headquarters, Burkina Faso's Information Minister Remis Fulgance Dandjinou confirmed on state TV.
There was no news of any Canadian casualties in the attack but the Canadian embassy in Ouagadougou urged all Canadians to remain indoors and follow instructions of authorities.
Soldiers, meanwhile, were guarding the army chief of staff's building, the prime minister's office and French Embassy.
Burkina Faso, like other countries in West Africa, has been targeted sporadically by jihadist groups operating across Africa's Sahel region.
In recent years, Al Qaeda in the Islamic Maghreb has pushed south from its base in Algeria, into Mali, and then to Burkina Faso.
In a post on Facebook, it said: "Attack under way at the French Embassy and French Institute".
It was the third major attack in Ouagadougou in just over two years.
Explosions, gunfire rock Burkina Faso's capital
But the French embassy themselves have come out to say they are not being targeted.
At the start of the Friday attacks, witnesses said, armed men got out of a auto and opened fire on passers-by before heading to the embassy.
That attack pivoted Burkina Faso as a natural target by a deadly cocktail of rebels and religious militias who continue to threaten much of West Africa, prompting the creation of a so-called G5 Sahel force to neutralize such a threat.
The neighborhood also houses other embassies, the prime minister's office and United Nations offices.
No group claimed responsibility for Friday's attacks.
In a separate development on Friday, the specialist USA website SITE, which monitors extremists 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.
With him gone, these trade-offs are bound to dry out, leaving those elements without any serious incentives to stop them becoming the country's destabilization force.
The force is made up of soldiers from Burkina Faso, Mauritania, Mali, Niger and Chad. An Australian doctor who had spent decades treating civilians was also abducted along this border and remains missing.