Betancourt, who grew up, studied and raised her children in Paris, said that she had "dreamed of this moment for seven years".
She praised the "extraordinary Colombian heroes" who had rescued her and also thanked Sarkozy for his efforts in securing her freedom.
The French president, who was visibly moved, said: "Ingrid Betancourt, welcome. France loves you."
Sarkozy held a reception for her at the presidential palace and huge posters reading "Free at Last" were unfurled on the building of the French National Assembly in her honour.
Barnaby Phillips, Al Jazeera's correspondent in Paris, said: "It's almost surreal, the journey that Betancourt has taken over the last 48 hours, literally from the most austere circumstances possible, in the remote jungles of Columbia.
"I think France plays an important role in her life, she was educated here, her first husband was French, her children are here.
"Also she has realised how ordinary French people have campaigned for her release. She spoke of how she listened whenever she could to French international radio, just how importance those messages were to her.
"I think she was reminded that French people were battling for her release.
Betancourt was among 15 captives rescued from the Revolutionary Armed Forces of Colombia (Farc) in an operation in which the Colombian military tricked the rebels into handing them over.
However, at a news conference on Friday, Betancourt was questioned about a Swiss radio report which said that Farc members were paid in order to secure her release.
The French-language station, quoting a source, said that the captives "were in reality bought for a high price, and the whole operation afterwards was a set-up".
It said that the United States, which had three of its citizens among those freed, was behind the deal, and put the price of the ransom at about $20m.
The report said that the wife of one of the guards was the go-between, having been arrested by the Colombian army. She was released to return to Farc, where she persuaded her husband to change sides.
Betancourt said that she doubted the report.
"Given what I saw during the operation - and frankly, honestly, I don't think I can be easily fooled - I don't believe that what I saw was a set-up," she said.
Betancourt appealed to Sarkozy for help in freeing thousands of other hostages who remain in Colombia's jungles.
"I still need you, because we cannot leave them where they are," she said at a party in her honour at the presidential palace. "They are suffering, they are alone."
The group is thought to hold about 30 foreign captives and hundreds more Colombian civilians and security forces members.
Sarkozy said that Betancourt's rescue sent a message to people in difficult situations that "it's worth it to fight. There is no such thing as inevitability".
"All those who suffer, like you, throughout the world should know that... there is a light at the end of the tunnel," he said in greeting her.
Hugo de Coulomme of the support committee for Ingrid Betancourt told Al Jazeera that he believed that there was much more to come from Betancourt.
"I think one has noticed from the way she has expressed herself from the moment she arrived in Colombia, in Bogota, in the airport, how determined she was and how concise she was about the future, about what she hoped to do for the other hostages."
On Saturday, she is expected to undergo medical tests at Val-de-Grace military hospital in Paris, Sarkozy's office said.
It said the chief doctor in the French president's office, Christophe Fernandez, had already given Betancourt a preliminary medical exam aboard the French government aeroplane that carried her to France.