Rock-throwing mobs confronted each other late on Friday after a cleric urged worshippers in his mosque to support the constitution, resulting in more than 15 people hurt. Police used tear gas after some protesters fired birdshot.
By early Saturday, calm had returned to the city, according to Khaled el-Azazi, spokesman for the regional security authorities. "We will arrest anyone who starts riots," he said.
Egypt's vote was staggered over two rounds because many judges refused to oversee the referendum, as is legally required.
Polls on Saturday opened in Cairo, Alexandria and eight other provinces and were scheduled to close at 9:00 pm (1900 GMT) after officials extended voting by two hours, the official MENA news agency reported.
Voting abroad in Egyptian embassies was also extended two days, to Monday.
The other half of the country is to vote on December 22.
Morsi cast his ballot at a polling station near the presidential palace in Cairo, state television showed. He made no comment to the media.
The president has ordered the military to help police maintain security until the results are known. A total of 130,000 police and 120,000 soldiers are being deployed, interior ministry and military officials told AFP.
Morsi's Muslim Brotherhood has thrown its formidable organisational machine behind a campaign in favour of the draft constitution.
"I'm voting for stability and for Dr Morsi's promised programme of renewal. I have gone over the text to compare it with what the opposition is saying, and what they say is false. It's a good constitution," said one Cairo voter, Enayat Sayyed Mostafa, a retired woman.
The main opposition coalition, the National Salvation Front, is urging a massive "no" vote to the draft charter.
Many of its supporters were especially hostile to the Muslim Brotherhood, which it believes wants to usher in sharia-style (strict Islamic) laws.
"I'm voting because I hate the Muslim Brotherhood, it's very simple. They are liars," said one voter, Abbas Abdelaziz, a 57-year-old accountant, outside a Cairo polling station.
Ali Mohammed Ali, an unemployed 65-year-old wearing a traditional long robe, said: "I voted for Morsi and it was a mistake, a big mistake. This constitution is bad, especially because it doesn't forbid child labour and opens the way for the marriage of minors."
International watchdogs, including the UN human rights chief, and the United States and European Union have expressed reservations about the draft because of loopholes that could be used to weaken human rights, including those of women, and the independence of the judiciary.
Analysts said it was likely -- but not certain -- the draft constitution would be adopted.
"If the constitutional draft is approved by less than 60 percent of the voters, it may become a linchpin for future disputation over the validity of the political system that springs from it," Yasser El-Shimy, a Middle East analyst for the International Crisis Group, told AFP.
"If more than 60 percent vote yes, it would be very hard for the opposition to claim that they represent the preferences of the Egyptian people at large," he said.