Traore said that order was being restored after violent protests against the French embassy and days of fighting as his faction moved to topple the government.
Splits have emerged within the army, with many soldiers appearing to seek Russian support as the influence of former colonial power France wanes.
At least three separate videos shared on online on Saturday and Sunday showed Burkina Faso soldiers atop armoured personnel carriers, waving Russian flags, while the crowd around chanted "Russia! Russia!". Reuters has not verified the videos.
Traore's team urged people to halt attacks on the French embassy, targeted by protesters after an officer said France had sheltered Damiba at a French military base in the West African country and that he was planning a counter-offensive.
The French foreign ministry denied the base had hosted Damiba after his ouster on Friday. Damiba also denied he was at the base, saying the reports were a deliberate manipulation of public opinion. His whereabouts remain unknown.
"We want to inform the population that the situation is under control and order is being restored," an army officer said in a statement broadcast on national television. Traore, an army captain, stood beside the officer and was flanked by other armed and masked soldiers.
Another statement said Traore would continue to act as president until the designation of a transitional civilian or military president in coming weeks.
Ouagadougou was mostly calm on Sunday after sporadic gunfire across the capital throughout Saturday between opposing army factions.
"We invite you to continue with your activities and refrain from all acts of violence and vandalism... notably that against the French embassy and the French military base," the officer loyal to Traore said, urging people to remain calm.
The soldiers announced that Burkina Faso's air borders had been reopened.
Damiba himself led a coup earlier this year against a civilian government that had lost support over rising violence by Islamist extremists. Damiba's failure to stop attacks by the militant groups had led to anger in the ranks of the armed forces in the former French-protectorate.
Divisions have emerged within the army also over whether to seek help from other international partners to combat the militants.
The soldiers who ousted Damiba said the former leader, whom they had helped to seize power in January, reneged on a plan to seek other partners.
They did not name the partners, but observers and supporters said the soldiers want closer partnership with Russia, as did the soldiers who seized power in neighbouring Mali in August 2020.
There are concerns that France's influence in its former colonies in the region is weakening, while Russia - riding growing anti-French sentiment - spreads its influence.
