Announcing the lifting of a curfew and the reopening of borders on Friday, Col Abdul Karimou told journalists the situation was now "under control".
"There is no single voice of dissension in either Niamey or in other parts of the country," he added.
Troops stormed the presidential palace during a cabinet meeting on Thursday afternoon, seizing Mr Tandja and his ministers before announcing that they were suspending the constitution and dissolving all state institutions.
Calling themselves the Supreme Council for the Restoration of Democracy (CSRD), the coup leaders promised to turn Niger into an example of "democracy and good governance" and save its people from "poverty, deception and corruption".
A senior army officer, Col Salou Djibo, was named head of a military government.
The US said it believed the coup had been triggered by Mr Tandja's actions last August, when he held a controversial referendum to abolish limits on presidential terms of office.
Niger is suspended from all activities of the AU. Meanwhile, we will continue with the process of helping them return to constitutional order
Mull Sebujja Katende, African Union Peace and Security Council
The BBC's Idy Baraou in Niamey says thousands of people took to the streets near the parliament building after Friday prayers in support of the coup.
The opposition Co-ordination of Democratic Forces for the Republic (CFDR) - a coalition of political parties, trade unions and human rights groups formed last year to protest against the constitutional changes - also welcomed it.
In a statement, the group condemned "President Tanja's stubbornness in power", and called on the junta to respect its pledge to restore democracy.
Mr Tandja, a 71-year-old former army officer, was first elected in 1999 and was returned to power in an election in 2004. His second term in office expired in December, but was extended to allow him to complete major investment projects.
'Benefit of the doubt'
The international community has been quick to condemn the coup.
The European Union followed former colonial power France in calling for all those involved to "engage immediately in a democratic process allowing for rapid establishment of the constitutional order".
Col Salou Djibo Coup leader
Col Djibrilla Hima Hamidou Involved in 1999 coup
Col Goukoye Abdul Karimou Junta spokesman
Col Amadou Harouna
The chairman of the African Union Commission, Jean Ping, said there should be "zero tolerance" of those taking power by force.
The regional organisation later announced that it had suspended Niger and called for its citizens to be allowed to elect the leader they wanted.
"Niger is suspended from all activities of the AU. Meanwhile, we will continue with the process of helping them return to constitutional order," said Mull Sebujja Katende, who chairs the AU Peace and Security Council.
The West Africa trade bloc, Ecowas, which had been mediating in talks between the government and opposition before the coup, has already been in touch with the junta and has dispatched a team to Niamey for talks.
Ecowas chief Mohammed Ibn Chambas told the BBC the soldiers could be given "the benefit of the doubt" if they moved quickly to restore civilian rule.
"With the events of the last few days, we need to make contact with the authorities to help them return the country quickly to constitutional government," he told Focus on Africa.
Our correspondent says the junta leader, Col Djibo, has command over about 40% of the uranium-rich country's military arsenal.
President Tandja's second term in office expired in December
The colonel's family has deep-rooted army connections, he adds.
The 49-year-old received training in Bouake, Ivory Coast, Morocco and China, military sources told the Reuters news agency. He also served with the UN peacekeeping forces in Ivory Coast and the Democratic Republic of Congo.
Another of the plotters, Col Djibrilla Hima Hamidou, was junta spokesman during the last military takeover in 1999.
The then military leader, Col Ibrahim Bare Mainassara, was assassinated during that coup, but civilian rule was restored within a year.
On Friday, the coup leaders issued a communique saying government departments would now be run by senior civil servants.
Niger has experienced long periods of military rule since independence from France in 1960.
It is one of the world's poorest countries, but Mr Tandja's supporters argue that his decade in power has brought a measure of economic stability.
Under his tenure, the French energy firm Areva has begun work on the world's second-biggest uranium mine - ploughing an estimated $1.5bn (£970m) into the project.
China National Petroleum Corporation signed a $5bn deal in 2008 to pump oil within three years.
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