"Let's go, Nico!" his supporters chanted until after midnight during party scenes in downtown Caracas. The vote was marked by low turnout.
The disputed victory is likely to heighten global pressure on Maduro.
The United States said it would not recognize the vote's results, while the European Union and some Latin nations warned before the vote that the election was shaping up to be unfair.
Turnout in the previous three presidential elections averaged around 79 percent.
Falcon's quick rejection of Sunday's election, and call for a new vote, was therefore a blow to the government's strategy.
Maduro, who faces a colossal task turning around Venezuela's moribund economy, has offered no specifics on changes to two decades of state-led policies.
A senior US official said Sunday the Trump administration might press ahead on threats to impose crippling oil sanctions. "I'm going to stubbornly and obsessively insist in dialogue for peace".
But he showed no sign of replaying Sunday's vote.
"If the opposition wants to do the same, they are free to do so", said Rigoberto Barazarte, the owner of a small auto wash business who wants to see a re-elected Maduro toughen his stance against elites he says are trying to sabotage Venezuela's economy. "We do not recognize this electoral process as valid", he told local media.More news: Opp Cong to meet Manipur Guv to stake claim to form govt
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"In the red points they were engaging in shameless vote manipulation", said Bertucci. I prefer to come here to get water rather than waste my time, ' said Raul Sanchez, filling a jug from a tap by a busy road in the arid north-western city of Punto Fijo because his community has not had running water for 26 days.
Maduro also called on his political rivals to join him for negotiations about the future of the country.
Costa Rica also expressed "its profound worry that the electoral journey did not take into account the participation of all of political actors, nor of global independent observers, which weakens democracy". But most "Red Points" were just a few steps away.
Luis Emilio Rondón, the one CNE member critical of the government, said he did not recognise the results because "Venezuelans' freedom to vote" had not been respected.
Presidential elections were held on 20 May.
"The people of Venezuela have made their pronouncement and we ask everyone, nationally and worldwide, to respect the results", she said.
Mr Falcon called for a new vote, complaining about the government's placing of almost 13,000 pro-government stands called "red spots" close to polling stations nationwide.
Many Venezuelans are disillusioned and angry over the election: They criticize Maduro for economic hardships and the opposition for its dysfunctional splits.
In response, many opposition leaders called for an election boycott.
Opinion polls say the overwhelming majority of Venezuelans distrust the electoral council.
Some anti-government activists said the opposition coalition should have fielded a candidate regardless of how uneven the playing field might be. "You seek treatment", said Nayra Martinez, a city employee in the wealthy Caracas district of Chacao who made a decision to buck her party's call to abstain. We're in the final phase of a tragic cycle for our country. "We have a president of the people!"