Ciolacu tended his resignation as leader of the Social Democrats on Monday morning.
Political commentator Radu Magdin said the support for Georgescu was unprecedented, with earlier opinions polls placing him at around 5% of the vote.
“Never in our 34 years of democracy have we seen such a surge compared to surveys,” Mr Magdin said.
The victory of Georgescu, who campaigned under the slogan “Restore the dignity of the Romanian nation”, has been warmly welcomed in Russian media.
He previously condemned the Nato ballistic missile defence shield based at the Deveselu military base in southern Romania as “a disgrace”.
Georgescu was a prominent figure in another far-right party, George Simion’s AUR (Alliance for the Union of Romanians), which eventually expelled him as too radical, after he praised leaders of the Iron Guard, a Romanian fascist movement during World War Two.
But late on Sunday night, George Simion endorsed Georgescu, and urged the 1.3 million voters who voted for him on Sunday to support Georgescu in the run-off.
Georgescu’s success poses a dilemma for the millions of Romanians who voted for other candidates.
If Elena Lasconi gets through to the second round, as expected, many supporters of the Social Democrats, especially in rural areas, would find it difficult to support such a liberal, progressive figure.
Campaigning in this election focused largely on the soaring cost of living, with Romania having the EU’s biggest share of people at risk of poverty. Resentment of handouts to Ukrainian refugees in Romania also played a role.
The president in Romania has a largely symbolic role but considerable influence on areas such as foreign policy.
Turnout was 51%, similar to the figure five years ago.