The inauguration ceremony at the Constitutional Court was broadcast nationally on Wednesday.
Addressing the nation on state television after taking the oath, Damiba paid tribute to the security forces and the country's population who he said have been facing threats from extremism for more than six years.
"I swear in front of the people of Burkina Faso and on my honour, to preserve and respect, to uphold and defend the constitution, fundamental acts and the law, to do everything to guarantee justice for all the inhabitants of Burkina Faso," he said.
"Our country has always been strong during storms. That's why these tough times must be for us an opportunity to get to better horizons."
Damiba seized power promising to secure the war-torn country from growing jihadi violence linked to al-Qaeda and the Islamic State group that's killed thousands and displaced more than 1.5 million people.
Since taking control, Damiba has met with security forces, civil society, diplomats and politicians, but hasn't set a timeline for the transition to elections, something the international community has said must be done as soon as possible.
When military regimes take power the return to democratic, civilian rule can be lengthy and uncertain, say conflict experts.
"Military regimes will often delay the transition back to civilian rule and will work to retain some form of political influence in the background in order to ascertain that their interests are maintained," said Alexandre Raymakers, senior Africa analyst at Verisk Maplecroft, a risk consultancy.
"Considering the deteriorating security situation in Burkina Faso, the new military-led government will likely argue that the security situation will not allow for a comprehensive electoral process to be held, thus delaying a return to civilian rule," he said.
Even though Damiba's had wide popular support since taking control, some locals say they don't support the Constitutional Council's decision to let him assume the presidency because he wasn't democratically elected.
"We don't understand this at all, how things can go like this? The Constitutional Council inaugurates presidents who come to power through elections ... It's like they're saying it's legal to have a coup in Burkina Faso," Oumar Cisse a resident in the Sahel's hard-hit Dori town told the AP by phone.
The international community has condemned the coup with the US halting financial assistance and the West African regional bloc known as ECOWAS and the African Union suspending Burkina Faso until former President Roch Marc Christian Kabore is released from house arrest.
Still most people, worried by the conflict with extremists, are hoping Damiba will be able to stem the violence. But the 41-year-old leader has yet to articulate a plan for how he will secure Burkina Faso better than the previous regime.
