The Islamist Muslim Brotherhood, which propelled President Mohamed Mursi to power in a June election, said an unofficial tally showed 64% of voters backed the charter after two rounds of voting that ended with a final ballot on Saturday.
An opposition official also said their count showed the result was a "yes" vote, while party spokesmen said there had been a series of abuses during the voting.
The main opposition coalition, the National Salvation Front, responded to the defeat by saying it was moving towards forming a single political party to challenge the Islamists who have dominated since Hosni Mubarak was overthrown two years ago.
Members of the opposition, taking heart from a low turnout of about 30% of voters, pledged to keep up pressure on Mr Mursi through peaceful protests and other democratic means.
"The referendum is not the end of the road," said Khaled Dawoud, a spokesman for the National Salvation Front. "It is only the beginning of a long struggle for Egypt's future."
The referendum committee may not declare official results for the two rounds until today, after hearing appeals. If the outcome is confirmed, a parliamentary election will follow in about two months.
Mr Mursi's Islamist backers say the constitution is vital for the transition to democracy, nearly two years after Mr Mubarak was ousted in a popular uprising, and that it will provide the stability needed to help a fragile economy.
The constitution is "a historic opportunity to unite all national powers on the basis of mutual respect and honest dialogue for the sake of stabilising the nation", the Brotherhood said in a statement.
The opposition accuses Mr Mursi of pushing through a text that favours Islamists and ignores the rights of Christians, who make up about 10% of the population, as well as women. They say it is a recipe for further unrest.
"The majority is not big and the minority is not small," liberal politician Amr Hamzawy said, adding that the National Salvation Front would use "all peaceful, democratic means" to challenge the constitution.
The vote was split over two days as many judges had refused to supervise the ballot, making a single day of voting impossible.
During the build-up to the vote there were deadly protests, sparked by Mr Mursi's decision to award himself extra powers in a November decree and then fast-track the constitutional vote.
The new basic law sets a limit of two four-year presidential terms. It says the principles of sharia, Islamic law, remain the main source of legislation but adds an article to explain this. It also says Islamic authorities will be consulted on sharia – a source of concern to Christians and others.
Rights groups reported what they said were illegalities in the vote. They said some polling stations opened late, that Islamists illegally campaigned at some polling places, and there were voter registration irregularities.
But the committee overseeing the vote said it detected no major irregularities in voting on December 15, which covered about half of Egypt's 51 million voters. About 25 million were eligible to vote in the second round.
Leader comment: Page 14