Mr Mansour said fresh elections were “the only way” forward, but gave no indication of when they would be held.
Mr Morsi, Egypt’s first freely elected leader, is under house arrest after what he says was a military coup.
The army said he had “failed to meet the demands of the people”.
The upheaval comes after days of mass rallies against Mr Morsi and the Islamist Muslim Brotherhood movement he comes from.
Protesters accused them of pursuing an Islamist agenda and of failing to tackle Egypt’s economic problems.
The health ministry says at least 10 people were killed and scores injured in clashes at rival protests across the country overnight. Some 50 people have died since the latest unrest began on Sunday.
Unnamed officials have said Mohammed Badie, supreme leader of the Muslim Brotherhood, has been arrested in Marsa Matrouh, a Mediterranean coastal city to the west of Cairo.
He is the latest high-profile detainee from the Brotherhood.
Mr Mansour was sworn in as chief justice of the Supreme Constitutional Court on Thursday morning, then, shortly after, he took the oath to become interim head of state, vowing to “preserve the system of the republic, and respect the constitution and law, and guard the people’s interests”.
He said he would safeguard “the spirit of the revolution” which removed Hosni Mubarak from power in 2011, and would “put an end to the idea of worshipping the leader”.
He held out an olive branch to the Muslim Brotherhood, saying they were “part of this people and are invited to participate in building the nation”.
Elections would be held based on “the genuine people’s will, not a fraudulent one,” he said. “This is the only way for a brighter future, a freer future, a more democratic one”.
Mr Mansour also praised the military as “the conscience of the nation and the guarantor of its security and safety”.
However Gehad el-Haddad, a spokesman for the Muslim Brotherhood, told the BBC that Mr Morsi had been put under house arrest and the “entire presidential team” was in detention.
Mr Haddad’s father, senior Morsi aide Essam el-Haddad, and Saad al-Katatni, head of the Brotherhood’s political wing, are among those being held, as well as supreme leader Mr Badie.
Continue reading the main story
We don’t have a democracy anymore. The military have taken it away from us. We will take to the streets and stay there”
Islam Hawala Giza
Voices from the streets
Arrest warrants have reportedly been issued for some 300 other members of the Muslim Brotherhood, including Mr Badie’s deputy, Khairat al-Shater.
The BBC’s Shaimaa Khalil in Cairo says the Brotherhood is refusing to acknowledge the change in power and is planning to hold rallies across the country.
Both sides are highly charged, our correspondent says, and confrontation on the streets appears inevitable.
Senior figures in the Brotherhood and its political wing, the Freedom and Justice Party, were quoted as saying they would not work with the new powers – but would not take up arms or encourage followers to do so either.
As Mr Mansour took his oath, military jets flew over the capital trailing the colours of the national flag.
Cairo’s Tahrir Square was calm on Thursday morning, though crowds were expected to gather later in the day.
Morsi supporters told BBC World Service that the army’s action did not mean the end of political Islam in Egypt
Many had celebrated in Tahrir Square through the night after the army announced on state TV Mr Morsi had failed to meet a deadline to respond to protesters’ demands.
The army’s roadmap for the post-Morsi era includes:
Suspension of the constitution
A civilian, transitional technocratic government
Supreme Constitutional Court to prepare for presidential and parliamentary elections
A “charter of honour” to be drawn up and followed by national media
Countries in the region have reacted cautiously to the events. Saudi Arabia, Qatar and the United Arab Emirates welcomed the change in leadership, while Turkey’s Deputy Prime Minister Bekir Bozdag cautioned that it was “not a result of the will of the people”.
Senior African Union officials said Egypt’s membership could well be suspended, as it would be in any country “where an unconstitutional change has taken place”.
US President Barack Obama said he was “deeply concerned” by the events, and called for a swift return to civilian rule, while UK Prime Minister David Cameron said a “democratic transition” must take place soon.
German Foreign Minister Guido Westerwelle described the ousting as “a serious setback for democracy” and called on all sides to refrain from “violence and arbitrariness”.
UN chief Ban Ki-moon noted the “legitimate concerns” of protesters, but said military inference was always “of concern”.
Meanwhile, Syrian President Bashar al-Assad welcomed events unreservedly, calling the ousting of Mr Morsi the “fall of so-called political Islam”.
The recent mass protests that led to the army’s intervention were called by the Tamarod (Rebel) movement, which said it had collected a 22-million strong petition demanding Mr Morsi stand down.
Mohammed Morsi became Egypt’s first Islamist president on 30 June 2012, after winning an election considered free and fair following the 2011 revolution that toppled Hosni Mubarak.
However his term in office was marred by constant political unrest and a sinking economy.
His moves to entrench Islamic laws and concentrate power in the hands of the Muslim Brotherhood also alienated liberals and secularists.
Are you in Egypt? What do you think about the removal of President Morsi? Have you been celebrating? What do you think about the current situation? Share your thoughts using the form below.