The result of the Tuesday-Wednesday vote had never been in doubt, as Morsi's Muslim Brotherhood and other Islamists boycotted it, but the authorities wanted a large turnout in the first democratic test since the ouster in July.
Army chief Abdel Fattah al-Sisi, the general who led the overthrow, had been monitoring the outcome for an indication of support for a possible presidential bid, military officials said.
Electoral commission head Nabil Salib said turnout "reached 38.6 percent" of 53 million registered voters, with only 1.9 percent voting "no."
That turnout proved "that the revolution of June 30 was a popular revolution," a senior government official said at a news conference to announce the results.
It "refuted the doubts of naysayers," said Salah el-Din Abdel Sadeq, head of the government's media arm.
The new charter replaces an Islamist-inspired one adopted in a December 2012 referendum under Morsi with about two-thirds of the vote and a 33 percent turnout.
The United States, a major source of financial aid most of which is military funding, reacted to the news by urging Egyptian authorities to fully implement the rights guaranteed under the constitution.
"It's not one vote that determines a democracy. It's what comes next that will shape Egypt's political, economic and social framework for generations," said US Secretary of State John Kerry.
"As Egypt's transition proceeds, the United States urges the interim Egyptian government to fully implement those rights and freedoms that are guaranteed in the new constitution for the benefit of the Egyptian people, and to take steps towards reconciliation," he said.
The US administration on Thursday said it was closely watching the results of Egypt's referendum, but had not yet decided whether to unfreeze some $1.5 billion (1.1 billion euros) in aid.
The military removed Morsi days after millions of protesters began rallying against him on June 30, setting off months of unrest by his followers who decried what they called a "coup."
Presidential and parliamentary elections have been promised for later this year.
'Sisi for presidency'
Sisi is wildly popular among the millions who took to the streets against Morsi, but the Islamist's followers revile him for what they say was a "coup" against Egypt's first freely elected and civilian president.
The now-banned Brotherhood, harried by a deadly crackdown since Morsi's removal, dismissed the referendum as "farce".
The group has called rallies for January 25, the third anniversary of the 2011 uprising that overthrew veteran strongman Hosni Mubarak.
Interior minister Mohamed Ibrahim also called for demonstrations the same day to counter an Islamist "plot to spark chaos," an unusual appeal from the top official tasked with enforcing a law that restricts protests.
More than 1,000 people, mostly Islamists, have been killed and thousands imprisoned in a police crackdown on pro-Morsi protests.
On Friday, four people were killed in fighting, the health ministry said, as police clamped down on the Islamist rallies.
Many who voted said their ballot was also an endorsement of Sisi, seen as a strong man capable of restoring security after the three years of turmoil following Mubarak's overthrow.
Morsi supporters want Sisi tried internationally for crimes against humanity over the deadly crackdown, but the general is adored by his supporters and will face no serious competition if he stands for election.
"If General Sisi nominates himself for president his chances will be great," presidential aide Ahmed al-Muslimani told the London-based Al-Sharq Al-Awsat newspaper in an interview.
Muslimani said Sisi has not yet made up his mind, but other officials say his candidacy appears to be a foregone conclusion.
In the first test of democracy after Morsi's overthrow, the run-up to the referendum was marred by arrests of activists who campaigned against the constitution.
"Egypt's government clearly doesn't realise that a 98% vote for anything virtually defines an unfair campaign," wrote Human Rights Watch's director Kenneth Roth on Twitter.
Democracy International said its monitors witnessed security forces and campaigning material inside polling stations, but there was "no evidence that such problems substantially affected the outcome of this referendum".
The vote has put the Brotherhood, which the government designated last month as a terrorist group, on the back foot.
Morsi himself has been in custody since his ouster and is currently standing trial in the first of three separate cases against him.