Those official results tallied with figures given by President Mohamed Morsi's Muslim Brotherhood immediately after the last round of polling over the weekend in the two-stage referendum. Turnout, however, was barely 33%.
"There is no loser in this referendum result. This constitution will be for all of us," Prime Minister Hisham Qandil said in a statement. He called on "all political forces to cooperate with the government" to restore the economy.
The opposition reiterated its rejection of the result.
"The law will take its course after the official complaints we have made to the prosecution service over violations and fraud that have been noted," National Salvation Front spokesman Khaled Daoud told AFP.
The main opposition coalition, however, has already dismissed the plebiscite as "only one battle" and vowed to "continue the fight for the Egyptian people".
That sets the scene for continued instability after more than a month of protests, including clashes on December 5 that killed eight people and injured hundreds.
Washington called on Morsi to work to "bridge divisions".
"President Morsi... has a special responsibility to move forward in a way that recognises the urgent need to bridge divisions, build trust, and broaden support for the political process," acting State Department spokesman Patrick Ventrell said in a statement.
EU foreign policy chief Catherine Ashton took note of both the majority backing the constitution and the low turnout.
Repeating a call for dialogue, she added: "I urge those concerned, in particular the president, to intensify efforts in this regard."
Egypt's communications minister Hany Mahmoud announced his resignation on Tuesday, following in the footsteps of Vice President Mahmud Mekki.
There have also been conflicting reports on whether central bank governor Faruq el Okda has stepped down, while the prosecutor general recently quit and then swiftly retracted his resignation.
"I quit for Egypt," the official Mena news agency quoted Mahmoud as saying, though the precise motives for his decision were not spelt out.
He apparently first tendered his resignation on November 22, the day Morsi announced sweeping new powers for himself.
Many creditors, investors and tourists have abandoned Egypt because of the volatility that has prevailed since the early 2011 revolution that toppled veteran leader Hosni Mubarak.
The International Monetary Fund this month put on hold a $4.8 billion loan Cairo needs to prevent a looming currency collapse.
Rating agency Standard and Poor's has downgraded Egypt's long-term credit rating one notch to "B-" because the "elevated" political tensions show no sign of abating.
Samir Abul Maati, president of the national electoral commission, told a Cairo news conference that a total of 63.8% of valid ballots supported the new constitution. Turnout was 32.9%.
Opposition allegations of fake judges supervising some of the polling were unfounded, he said, though adding that results from a few polling stations were annulled because they had closed early.
The opposition has seized on irregularities and the low turnout to challenge the legitimacy of the charter.
Front leader Mohamed ElBaradei, a Nobel Peace laureate and former chief of the UN atomic energy agency, however conceded that the referendum was going to pass.
"But it's a really sad day in my view for Egypt, because it is going to institutionalise instability," he told the US television network PBS on Monday.
ElBaradei said the new charter should be treated as an "interim one" until another is written up on the basis of consensus.
The opposition argues that the Muslim brotherhood and Salafist groups want to use some of the charter's ambiguous language to slip in sharia-style strict Islamic law.
Both the opposition and UN human rights chief Navi Pillay have criticised the text, written by a panel dominated by Islamists, for weakening women's rights and other rights, including those of non-Muslims.
The brotherhood argues that the constitution is a necessary step to restoring stability.
Its religious leader, Mohamed Badie, tweeted: "Congratulations to the Egyptian people on approving the constitution of revolutionary Egypt. Let's start building our country's rebirth... men and women, Muslims and Christians."
The low turnout, though, confounded the brotherhood's public predictions that voters would give it greater support.
"Anything less than 70% would not be good," Amr Darrag, a senior member of the Brotherhood's Freedom and Justice Party told AFP on December 2. He helped draft the constitution.
Attention now turns to legislative elections that must be held by the end of February. The constitutional court dissolved the previous parliament in June.
Morsi has ordered the senate, which currently handles all legislative business, to convene on Wednesday, the official MENA news agency said.