The attack is now over, authorities say, with at least two assailants killed by security forces.
A jihadist attack on a nearby cafe killed 30 people in January last year.
There are fears that the latest attack is the work of one of the affiliates of al-Qaeda that are active in the Sahel region, the BBC's Alex Duval Smith reports.
Burkina Faso's new war against militant Islamists
Minister of Communication Remis Dandjinou confirmed on Monday that the attack was over, with security forces carrying out checks on neighbouring buildings.
The shooting began shortly after 21:00 (21:00 GMT) on Sunday on Ouagadougou's busy Kwame Nkrumah Avenue.
The Aziz Istanbul Restaurant appears to have been at the centre of the attack.
One eyewitness told the BBC: "I saw there were multiple trucks or jeeps driving through my street, with... local army/police officers with AK47s, deploying in front of my house.
"I heard a lot of shootings and then I was scared as hell and I went inside. I've been hearing quite a bit of shooting."
Police captain Guy Ye told Associated Press that the attackers had arrived on motorcycles and had begun shooting randomly.
It was not clear if the death toll of 18 included the two attackers.
Turkey's foreign ministry has confirmed that one of the dead was Turkish. Unconfirmed reports say another was French.
The attack is similar to one in January 2016, when gunmen targeted the Splendid Hotel and the Cappuccino restaurant, only 200m further along Kwame Nkrumah Avenue from the scene of the latest attack.
More than 170 people were taken hostage and 30 were killed. The al-Qaeda in the Islamic Maghreb (AQIM) group said it carried out that attack.
Burkina Faso is part of the Sahel region, which includes Mali where Islamist groups have been active since 2012.
A multinational force run by African nations to target jihadist forces in the Sahel region has been established, but it will not be operational until later this year.