Georgescu won 23% of the vote, with 19% for the runner up, Elena Lasconi, of the opposition Save Romania Union.
Prime Minister Marcel Ciolacu of the governing Social Democrats came third.
The Central Election Bureau must now decide how to carry out the recount, with what personnel, and by what deadline.
A full recount is unprecedented in Romanian post-Communist history.
As things stand, Georgescu is due to face Lasconi in a run-off on 8 December.
“Extremism is fought by voting, not backstage games,” Lasconi said.
“I call on the Central Election Bureau to handle the vote recount wisely. The law must be the same for all, not interpreted differently for some.”
TikTok also faced accusations that it did not respect electoral rules by Romania’s top security body, the Supreme Council of National Defence.
Outgoing President Klaus Iohannis, who convened the council, said the platform “did not mark him as a political candidate”.
But TikTok has vehemently denies the allegations.
“It is categorically false to claim his account was treated differently to any other candidate,” it said in a statement.
“When Romanian authorities contacted us to flag a number of videos that lacked identifiers… we took action on those videos within 24 hours.”
Georgescu, himself, has pushed back against criticism that he used the social media platform illegally to gain an electoral advantage.
The 62-year-old has more than 330,000 followers – up from 30,000 just over a fortnight ago – and more than 4m likes.
“The budget of this campaign was zero… I had a very small team – a maximum of 10 people, no more. But we had millions of people behind [us],” he told the BBC.
“I’m not different – the Romanian people are different. Romanian people need freedom. Real democracy means spirituality. God. Our land. Our property. Our soul. Our family.”
He added that state institutions were trying to deny the people’s choice.
Anti-Georgescu protesters have already taken to the streets in Bucharest and several provincial cities, while Georgescu has appealed to his supporters to “stay home with friends and family” and not respond to provocations.
Romania’s Telecoms regulator Ancom has called for TikTok to be suspended pending an investigation by prosecutors into suspicions of manipulation of the electoral process.
Romania’s National Audiovisual Council has also asked the European Commission to investigate the way TikTok, which bans formal political advertising, was used in the election.
Romania will hold a parliamentary election this weekend, with the far-right parties AUR and SOS Romania hoping for a surge in popularity in the wake of the presidential vote.
The parties of the governing coalition, Social Democrats and National Liberals, are in disarray – humiliated by the failure of their candidates in the presidential election.
Across Romania, and in the large Romanian diaspora abroad, there’s a mood of elation, despair, or simply confusion.