According to the opposition, at least 16 people were killed during anti-government protests on Saturday and Saturday.
As voting got under way on Sunday morning, anti-government protesters took to the streets despite a government ban and there were reports of clashes with police across the country. Authorities said 10 people died in the various protests and the opposition said the true death toll was 15.
"The first vote for peace", the Venezuelan president said after casting his ballot. "Today, the Venezuelan people won".
The vote capped a week of tensions in Caracas, where numerous city's 2 million residents stockpiled food and water amid a two-day national strike. "It's the biggest vote the Bolivarian revolution has had in 18 years".
The turnout during the election was 41.5%, the country's Election Commission President Tibisay Lucena said.
Speaking at a press briefing in Caracas, Capriles said voter participation Sunday was less than 15%. Around 380,000 troops guarded the voting centers as people stepped out to cast their votes.
"Tomorrow (Monday), we will start the ANC and, through it, the path to recovering the guarantees of peace and dialogue among all Venezuelans.hopefully the opposition understands this", he added.
In Sunday's gravest incident, a bomb went off as a group of police officers on motorbikes sped past Caracas' Altamira Plaza, an opposition stronghold.
Among those killed was was a 39-year-old lawyer in the southeastern town of Ciudad Bolivar who was a candidate to sit on the new assembly. State prosecutors are investigating both deaths.
Among the elected members are the wife of President Maduro, Cilia Flores, and the second man of Chavismo, Diosdado Cabello.
He praised the "massive" turnout for the election, despite "provocations" by the opposition.
Months of opposition to President Nicolas Maduro's plan to strengthen his party's power has resulted in more fatal clashes on the day of the election.
Congress has already been severely weakened by the Maduro-loyalist electoral commission and supreme court.
Speaking after Sunday's controversial election, Maduro claimed that the opposition turned down an offer from his government to postpone the vote.
The people of Venezuela voted to create the National Constituent Assembly consisting of delegates who would rewrite the country's constitution.
Opposition leaders had called for a boycott of the vote, declaring it rigged for the ruling party.
For the president, the hand-picked successor to Hugo Chavez, the creation of the so-called constituyente was another step in a stubborn rise to autocratic power in the face of global condemnation, USA sanctions, a cratering economy and months of civil unrest that has claimed more than 110 lives. But critics say it will erode democracy.
He has pressed ahead with the vote to create the all-powerful constituent assembly or temporary parliament despite U.S. sanctions and months of violence which has claimed the lives of around 120 people.