Calvo did not speak at a news conference, but gave a thumbs up sign.
Afterwards, the soldier and his family were flown to the capital, Bogota, where Calvo was treated at the Military hospital for dehydration and was in stable condition, its director said in a statement.
Colonel Nora Ines Rodriguez added that three gunshot wounds suffered a year ago in his right leg had healed but a fourth had damaged the top of his left knee.
'Peace is possible'
Piedad Cordoba, the opposition senator who led the rescue mission, said Calvo was emotional and lightheaded during the flight from the village of Santa Lucia, where rebels handed him over.
|Calvo was injured and captured while in combat last April [AFP]
"Yes, peace is possible, it's irreversible," Cordoba said in comments on the release that she has been running on her Twitter feed.
Calvo is the first of two soldiers the Farc said it would free this week in what the rebels call their last goodwill unilateral release.
The other soldier that Farc has said it will release as what the rebels have said will be their last goodwill release is Sergeant Pablo Emilio Moncayo.
The 32-year-old who has been held for more than 12 years and his father has gained fame for walking halfway across Colombia to press for his release.
Moncayo is expected to be freed on Tuesday.
Farc says it will now demand a swap of jailed rebels in exchange for the 20 police and soldiers it still holds, most of whom have been held for more than a decade.
Alvaro Uribe, the Colombian president who steps down in August after two consecutive four-year terms, is popular in Colombia for aggressively fighting Farc and dealing it significant blows, including the 2008 rescue of former presidential candidate Ingrid Betancourt, three US military contractors and 11 other captives.
Swap in doubt
Speaking in the town of Arauca, Uribe welcomed Calvo's release and said he spoke with the soldier by telephone. He said he does not object to prisoner exchanges as long as they do not return "criminals to the Farc".
But Uribe has always insisted that any rebels freed from Colombia's prisons as part of a prisoner swap be taken by another country, such as France.
He has called the Farc's unilateral releases cynical publicity stunts that belie their disregard for civilian life.
Just last Wednesday, a bombing in the administrative centre of the Pacific port of Buenaventura that killed nine people, was attributed to Farc.
Analysts say they expect no movement on a possible prisoner swap until a new president takes office on August 7 after elections, the first round of which are set for May 30.
Los Andes University political scientist Sandra Borda noted that none of the presidential candidates have said they are disposed to negotiate with the Farc, which she said is "weakened but far from being defeated militarily".
Calvo is the first Farc captive to be released since February 2009, when the guerrillas handed over two politicians, three police officers and a soldier.