The symbolic "plebiscite" was taking place after nearly four months of violent protests against Maduro, during which nearly 100 people have died.
It laid a direct challenge to a controversial plan by the president to have a citizens' body elected on July 30 to revise the constitution.
Detractors say that move is aimed at giving Maduro dictatorial powers to hold on to office, notably by bypassing the opposition-controlled parliament.
Maduro, though, says the new body, called the "National Constituent Assembly," is the only way to lift Venezuela out of its economic and political crisis.
"We are moving towards a Constituent Assembly to save the nation from attacks by fascists, imperialists and violence," he said.
In each of the votes, however, the other side is refusing to take part.
International unease
The opposition has presented its vote as the only way to counter Maduro's scheme.
"Maduro wants to make Venezuela into Cuba, and we can't let that happen. That's why on Sunday we all have to come out and vote," parliamentary leader Julio Borges said.
The opposition coalition, the MUD, is organizing Sunday's vote because the National Electoral Council -- which has systematically sided with Maduro -- has refused to authorize it.
But the Organization of American States has given its backing, and five former Latin American presidents have been invited in as observers.
The same day the government is to steal some of the opposition's thunder by holding a dry-run of the July 30 election, which the electoral council has okayed.
The head of the electoral council, Tibisay Lucena, has dismissed the opposition vote as just one more protest.
Given the possibility of the rival campaigns getting out of hand, there are rising calls for the opposition and government to open dialogue.
UN Secretary General Antonio Guterres said talks were "urgently" needed to stem the violence.
"Only a political solution can restore hope to Venezuela," he said in a statement.
Maduro's bid for a Constituent Assembly has been slammed not only by the opposition but also Venezuelan Attorney General Luisa Ortega -- who has dramatically broken ranks with Maduro -- and the Catholic Church.
Criticisms include the fact that no referendum was held before Maduro decided to go ahead with the new body, and that its election was being done through choices made by sectors and territories that can be influenced by the government.
Disaffected voters
The MUD coalition has called for public disobedience against the "dictatorship" and portrayed its vote on Sunday as a trigger that could end up forcing Maduro from power through massive protests or a general strike.
Its campaign has found echo in the scarcities suffered by many Venezuelans.
"I'm sick of not managing to buy food or medicine, with my insufficient salary," a 29-year-old nurse, her face hidden by a scarf during a protest, told AFP.
But Maduro still enjoys loyalty from electoral and judicial institutions, and -- most importantly -- from his military chiefs.
The changes he is looking for would allow him to override the parliament and do away with his rebellious attorney general.
He also wants more control over social programs, food distribution and price controls to rein in inflation -- which the International Monetary Fund estimates will be 720 percent this year.
While around 80 percent of Venezuelans disapprove of Maduro's performance, according to Luis Vicente Leon of the Datanalisis firm, the opposition's challenge is to use its vote "as a motivator for peaceful protests able to topple the government."
Yet Maduro faces his own challenge with the July 30 vote. Around 70 percent of voters reject his Constituent Assembly.
An absence of legitimacy there, coupled with the fracturing of the socialist "revolution" he inherited from late president Hugo Chavez and international isolation mean his authority would become more and more limited, Leon said.
Abstention would be a key metric in that vote, said another analyst, Benigno Alarcon.
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