Araujo was kidnapped while jogging in the Caribbean resort city of Cartagena in December 2000. He had been Andres Pastrana's, the former president, development minister.
The military operation, in which a soldier and six guerrillas were killed, came less than three months after Alvaro Uribe, the Colombian president, ordered the army to rescue hostages.
The order followed the October explosion of a car bomb in the parking lot of a military university in Bogota, the Colombian capital, which was blamed on the FARC.
Farc says it fights to close the gap between rich and poor in the Andean country.
"This is an important success for the government, but their are still plenty of other hostages out there," said Ricardo Avila, a political commentator.
They include Ingrid Betancourt, who was captured during her 2002 presidential campaign, and three US defence contractors kidnapped in 2003.