She told reporters she was "so proud" of her children, who were 16 and 13 when she was seized.
"They fought a battle, they dug deep into their characters, from the depths of their souls and fought a very beautiful battle," she said.
She described the reunion as "an orgy of kisses".
"I don't know what I said to them. I think I cried... I told them that they would have to put up with me because I was going to stick to them like bubblegum and I was going to eat them up with kisses."
Her daughter said the reunion was "the toughest and most beautiful moment that my brother and I have lived through".
Earlier, Ms Betancourt had hailed Wednesday's mission to free her and 14 fellow hostages as a "perfect operation".
Colombian soldiers, posing as members of a non-government organisation, had flown the hostages to freedom in a helicopter.
The local commander in charge of the hostages, a man known as Cesar, and another rebel were captured in the operation and will face justice, the authorities say.
Also released were three Americans and 11 members of the Colombian security forces.
The freed Americans - military contractors Marc Gonsalves, Thomas Howes and Keith Stansell - were flown to Texas late on Wednesday, and the US military later said they all were "in a very good physical health".
The military said the three men - who had been held by Farc since February 2003 - would now undergo medical tests and would be reunited with their families at "a private location".
Feb, 2002: Betancourt kidnapped by Farc rebels
Feb, 2003: US defence contractors Thomas Howes, Keith Stansell and Marc Gonsalves seized by after their plane goes down in southern Colombia
Jan, 2008: Betancourt aide Clara Rojas and ex-congresswoman Consuelo Gonzalez freed by Farc
March, 2008: Colombian forces raid rebel camp in Ecuador and kill Farc commander Raul Reyes
March, 2008: Farc leader Manuel Marulanda dies of reported heart attack
July, 2008: Colombian military frees Ms Betancourt, the three US contractors and 11 other hostages
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