The election "without any doubt lacks legitimacy and we categorically refuse to recognise this process", Mr Falcon told supporters minutes before the results were announced.
The former secretary general of the Spanish Socialist Workers Party said: "tomorrow we have a government in Venezuela, an opposition, some very important economic problems, social problems that will surely need changes by of all".
Voting centers across Venezuela appeared largely empty for the election despite assurances from government officials that millions had turned out to vote by midmorning.
Maduro, however, dismissed such criticisms in his late-night victory address. Mr Bertucci, a TV evangelist who handed out soup at his campaign rallies, stopped short of challenging the results, partly blaming what he called a mistaken opposition boycott that led to a turnout of around 46% - the lowest in a presidential race in two decades of revolution.
"The whole of Venezuela has triumphed! Constitutionality has triumphed [These were] elections that were constitutional, legitimate and legal", he said. "Venezuela is very sick and we the people are the medicine".
The country's main opposition coalition, the Democratic Unity Roundtable had boycotted the election, while the two opposition leaders, Henrique Capriles and Leopoldo Lopez, were barred from contesting the elections.
According to The Associated Press, "Falcon was joined in his call for a new election by third-place finisher Javier Bertucci, who got around 11 percent of the vote".
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Most Venezuelans gave up long ago on "elections" as the way to restore their rights.
According to reports, the ballots were recorded electronically, making the voting quick and easy. Some 150 global observers from 30 countries and worldwide organizations were present in the Venezuela to monitor the whole election process.
Australia urged all Venezuelan political parties to seek a solution to the country's crisis through dialogue.
Hundreds of Venezuelans took to the streets in several Latin American capitals, including Bogota, Buenos Aires and Lima-as well as in Madrid-to denounce the vote.
Venezuela held what it called presidential and state legislative "elections" on Sunday, as if voters' ballots would decide who governs them.
US President Donald Trump has said in the past he would not rule out military action against Venezuela, which is home to 30 million people.
Information Minister Jorge Rodriguez said about 2.5 million people had voted by 10 a.m.
"It is an advanced automatic voting system".
The officials said the order prohibits all transactions related to the purchase of any debt owed to the Venezuelan government by any U.S. person or anyone within the U.S. The officials said it included any debt associated with Venezuela's state oil company, PDVSA.