The upheaval has raised concerns about the security of a region where Niger has been a key ally of Western powers seeking to contain insurgencies by groups linked to al Qaeda and Islamic State.
Tiani was the head of the presidential guard whose soldiers shut Bazoum inside his palace on Wednesday, leaving confusion over who was in control.
The general appeared on state television on Friday with a banner on the screen that described him as the president of a newly formed military body, the National Council for Safeguarding the Homeland (CNSP).
"The President of the CNSP is the head of state," an officer said, reading out a statement.
The constitution has been suspended, all government institutions dissolved and the CNSP will exercise all legislative and executive power until constitutional order returns, the statement added. It gave no timelines.
African countries, Western powers, regional and international organizations have reacted with alarm, insisting that Bazoum be freed and democracy restored.
One of the few to welcome the takeover was Wagner mercenary boss Yevgeny Prigozhin who remains active despite leading a failed mutiny against the Russian army's top brass last month.
He described the coup as an uprising against colonizers and offered his fighters' services to bring order.
Russian foreign minister Sergei Lavrov said constitutional order should be restored.
Niger is the world's seventh-biggest producer of uranium, the radioactive metal widely used for nuclear energy and in nuclear weapons, as well as for treating cancer.
Like the military rulers of Mali and Burkina Faso, Tiani justified the coup by saying the government had been failing to contain the Islamist insurgency.
Militants linked to al Qaeda and Islamic State have been spreading across West Africa's Sahel region for years. Niger so far has held them off better than Mali and Burkina Faso, where violence has only worsened since the military takeovers.
The juntas in Mali and Burkina Faso have increasingly turned toward Russia as a strategic ally and distanced themselves from traditional partners such as France, which has faced a growing wave of resentment towards its influence in the Sahel.
Bazoum has not made a statement since Thursday morning, when he vowed to protect "hard-won" democratic gains in a post on social media.
Several world leaders said that they have spoken to him since the coup, and that he is being detained with his family in the presidential palace but is "fine".
The United Nations said it would still deliver aid in Niger even though it had not had any contact with the military since the coup.
