Venezuela calls the move an effort to free the socialist-run country from the "tyranny of the dollar," echoing a plan recently announced by President Nicolas Maduro.
The opposition, which controls the legislature, has been sidelined and its leaders are under threat of arrest after organising protests.
The sanctions freeze any assets Maduro may have in US jurisdictions and bar Americans from doing business with him. Maduro says the sanctions reflect Trump's "desperation" and "hatred" for Venezuela's socialist government.
Opposition leaders had called for a boycott of the vote, saying it was rigged in favour of the ruling party, and by late afternoon they were declaring the apparent low turnout to be a resounding victory.
The opposition says Maduro's planned Constituent Assembly is a farce designed purely to keep him in power.
In their bid to pressure Maduro into abandoning a weekend election for a super-congress, millions of Venezuelans are expected to take part in a nationwide strike from Wednesday.
The Democratic Unity coalition had called a march on the pro-Maduro Supreme Court in support of alternative magistrates appointed by the opposition, but security forces blocked their way with armoured cars and riot shields.
The unofficial referendum will take place two weeks before President Nicolas Maduro is asking Venezuelans to go to the polls to choose delegates for a special assembly to rewrite the constitution.
Victory for Albania's Socialists empowers them to push economic and judicial reforms vital for membership of the European Union.
The exit poll shows the Socialists are set to win between 45 and 49 percent of the vote.
Venezuela's foreign minister blasts some members of the 34-nation Organization of American States after they chastise the current situation in the Latin American country.
Opposition legislators have taken their message onto public transport. On the streets, two more students died in the latest anti-Maduro demonstrations. The death toll since the start of the protests in April is now more than 70.
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