Moncayo was generally in good health, Adolfo Beteta, a spokesman for the International Red Cross, said.
The sergeant was 19 when he was taken captive during a rebel attack on an army outpost in the mountains on December 21, 1997. Libio Jose Martinez, a 33-year-old sergeant who was captured at the same time, is still being held.
Moncayo was one of the longest-held hostages and had only been seen occasionally in rebel videos, becoming a symbol of those left behind in the waning war against Latin America's longest-running insurgency.
Moncayo's father, Gustavo Moncayo, gained fame for walking halfway across Colombia in 2007 to press for his son's release. He has also lobbied governments from Venezuela to France, often wearing chains he said symbolised his son's captivity.
On Tuesday, Moncayo symbolically pulled off the chains from his father's hands.
Al Jazeera's Latin American editor, Lucia Newman, said Moncayo thanked the presidents of Ecuador, Venezuela and Brazil, saying they helped secure his freedom, but conspicuously left out any mention of his own president, Alvaro Uribe.
Uribe, who has previously called Farc's unilateral releases publicity stunts, welcomed Moncayo's release and thanked Brazil, the Red Cross and the Roman Catholic church for their co-operation.
"Colombia receives those who return from captivity with open arms and rejects the kidnappers with the greatest strength," he said in a statement.
The sergeant's release on Tuesday was the second this week by Farc, once a powerful rebel force that has been hobbled by Colombia's war on guerrillas and cocaine traffickers
The rebels on Sunday freed Josue Daniel Calvo, who was taken captive a year ago after he was wounded in combat.
The handovers come before Colombians go to the polls in May to pick a successor to Uribe, who steps down after two terms dominated by his hard-line campaign against Farc, which still holds about 50 police officers, soldiers and civilians.
Cordoba, an opposition senator who has been a go-between in contacting Farc, relayed a message from the rebels saying there would be no more unilateral releases after Moncayo.
The releases have reopened discussions about a possible broader agreement to negotiate an exchange of jailed rebels for kidnapped troops. But past hostage releases have not led to any such agreement or opened up peace talks.
|Moncayo appeared to be in good health as he embraced family members [EPA]
Uribe, whose father was killed in a botched rebel kidnap bid two decades ago, says he is open to an exchange if freed rebels do not return to crime and if the handover does not mean demilitarising an area that would allow rebels to regroup.
Farc has in the past demanded Uribe pull troops back from a zone the size of New York City to guarantee any handover. They had also wanted to include several extradited Farc leaders held in US jails in any swap.
Still, their recent communiques have not mentioned these conditions.
But broader peace talks to end the four-decade insurgency appear unlikely with Uribe, who demands the rebels cease hostilities before any talks can begin. Any candidate to replace Uribe in this year's election is likely to maintain his popular, tough line with the guerrillas.
Once an army that bombed and kidnapped at will, Farc has lost top commanders and seen its ranks thinned by desertions.
It is still, however, a threat in rural areas where state presence is weak, thanks to cash from cocaine trafficking.