Authorities said seven people were killed in the confrontations. The opposition said the true death toll was around 12, which would make Sunday one of the deadliest days since massive and sustained protests started in early April.
Maduro, widely disliked for overseeing an unraveling of Venezuela's economy, has promised the assembly will bring peace by way of a new constitution after four months of opposition protests in which about 120 people have been killed.
But opposition parties sat out the election, saying it was rigged to increase Maduro's powers, a view shared by governments including those of Colombia, Argentina and the United States.
U.S. Ambassador to the United Nations Nikki Haley tweeted "Maduro's sham election is another step toward dictatorship. We won't accept an illegit govt."
Caracas was largely shut down with deserted streets and polling stations were mostly empty, dealing a blow to the legitimacy of the vote. The government nonetheless extended voting by an hour to 7 p.m. EST (2300 GMT), citing what it described as long lines at voting centers.
Critics say the assembly will allow Maduro to dissolve the opposition-run Congress, delay future elections and rewrite electoral rules to prevent the socialists from being voted out of power in the once-thriving OPEC nation.
In what could be the spread of more aggressive tactics, a bomb exploded in Caracas and injured seven police officers.
The opposition has vowed to redouble its resistance and U.S. President Donald Trump has promised broader economic sanctions against Venezuela after the vote, suggesting the oil-rich nation's crisis is set to escalate further.
"Even if they win today, this won't last long," said opposition supporter Berta Hernandez, a 60-year-old doctor, in a wealthier Caracas district. "I'll continue on the streets because, not long from now, this will come to an end."
Maduro, a former bus driver and union leader narrowly elected in 2013, dismisses criticism of the assembly as right-wing propaganda aimed at sabotaging the brand of socialism created by his mentor and predecessor, the late Hugo Chavez.
"The 'emperor' Donald Trump wanted to halt the Venezuelan people's right to vote," said Maduro, as he rapidly voted at 6 a.m. in a low-income area of the capital Caracas that has turned on the government.
"A new era of combat will begin. We're going all out with this constituent assembly."
But with polls showing some 70 percent of Venezuelans oppose the vote, the country's 2.8 million state employees are under huge pressure to participate — with some two dozen sources telling Reuters they were being threatened with dismissal.
Workers were being blasted with text messages and phone calls asking them to vote and report back after doing so.
The opposition estimated participation was at around a paltry 7 percent by mid-afternoon, but warned the government was gearing up to announce some 8.5 million people had voted.