Queues formed at polling stations in Cairo and other cities while soldiers joined police to secure the referendum process after deadly protests in the run-up to the vote.
Egypt's President Mohamed Mursi provoked angry protests when he issued a decree last month expanding his powers, then fast-tracked the draft constitution through an assembly dominated by his Muslim Brotherhood group and its allies. At least eight people were killed in clashes last week outside the president's palace.
The liberal, secular and Christian opposition says the constitution is too Islamist and tramples on minority rights. Mursi's supporters say the charter is needed if progress is to be made towards democracy nearly two years after the fall of military-backed strongman Hosni Mubarak.
"The sheikhs told us to say 'yes' and I have read the constitution and I liked it," said Adel Imam, 53, who was queuing to vote in a Cairo suburb. "The president's authorities are less than before. He can't be a dictator."
Official results will not be announced until after a second round of voting next Saturday. But partial results and unofficial tallies are likely to emerge soon after the first round, giving a general trend.
In Alexandria on Friday, tensions boiled over into a brawl between rival factions armed with clubs, knives and swords. Cars were set on fire and a Muslim preacher who had urged people to vote "yes" to the constitution was trapped inside his mosque by opposition supporters.
Christians, who make up about 10% of Egypt's 83 million people and who have long grumbled of discrimination, were among those queuing at a polling station in the port city to oppose the basic law. They fear Islamists, long repressed by Mubarak, will restrict social and other freedoms.
"I voted 'no' to the constitution out of patriotic duty. The constitution does not represent all Egyptians," said Michael Nour, 45, a Christian schoolteacher in Alexandria.
Islamists are counting on their supporters and the many Egyptians who may fall into line in a bid to end the turmoil that has hammered the economy and sent Egypt's pound to eight-year lows against the dollar.
"I voted 'yes' for stability," said shopkeeper Ahmed Abou Rabu, 39. "I cannot say all the articles of the constitution are perfect but I am voting for a way forward."
Mursi was an early voter after polls opened at 8am. He was shown on television casting his ballot then dipping his finger in ink, a measure to prevent people voting twice.
Polls had been due to close at 7pm, but this was extended to 11pm.
After weeks of turbulence, there has been limited public campaigning. Disparate opposition politicians and parties, beaten in two elections since Mubarak's overthrow, only announced on Wednesday they backed a "no" vote over a boycott.
If the constitution is voted down, a new assembly will have to be formed to draft a revised version, a process that could take up to nine months.
Just over half of the electorate of 51 million will vote in the first round in Cairo and other cities. The army has deployed about 120,000 troops and 6000 vehicles to protect polling stations and government buildings.