The presidency announced that months of talks with the jihadists had "yielded results".
The presidency announced late on Saturday that months of talks with the jihadists had "yielded results" some six months after 21 of their classmates were freed with the help of international mediators.
"Today 82 more Chibok girls were released," a statement said.
"After lengthy negotiations, our security agencies have taken back these girls, in exchange for some Boko Haram suspects held by the authorities."
No details were given about how many suspects were released or their identities.
But AFP understands three Chadian nationals, allegedly senior commanders under Boko Haram leader Abubakar Shekau, were handed over.
Military and civilian militia sources in the town of Banki, on the border with Cameroon, said the girls left for Borno state capital Maiduguri on board six military helicopters at 6:10 am (0510 GMT).
"One of the girls was carrying a baby with her, a boy of less than two years," said the source on condition of anonymity.
The presidency said the teenagers would be brought to Abuja to meet Buhari, who was swept to power on a promise to defeat Boko Haram, whose insurgency has killed at least 20,000 people in Nigeria since 2009.
Shehu Sani, a Nigerian senator who has been involved in previous negotiations, said the talks lasted for "three to four months".
The government would now look to secure the release of the remaining hostages, he added.
Boko Haram fighters stormed the Government Girls Secondary School in the remote town of Chibok on the evening of April 14, 2014 and kidnapped 276 teenaged girls who were preparing to sit high school exams.
Fifty-seven managed to escape in the hours that followed but the remaining 219 were held by the group.
Boko Haram's Shekau claimed in a video message that they had converted to Islam.
The audacious kidnapping brought the insurgency to world attention, triggering global outrage that galvanised support from the former US first lady Michelle Obama and Hollywood stars.
The girls have become a symbol of the conflict. Last month, parents and supporters marked the three-year anniversary of the abduction, describing the situation as an unending "nightmare".
But they said previous releases had given them strength.
Enoch Mark, a Christian pastor whose two daughters were among those kidnapped, said of the latest releases: "This is good news to us. We have been waiting for this day.
"We hope the remaining girls will soon be released."
Patrick Yousef, the deputy regional director for Africa at the International Committee of the Red Cross (ICRC), confirmed the group "facilitated the safe return" of the girls as a "neutral intermediary".
The Swiss government was also involved, Nigeria said.
Military and civilian militia sources in Banki said the girls were brought back to the town in ICRC vehicles late on Saturday afternoon and stayed in the military barracks there overnight.
Boko Haram has used kidnapping as a weapon of war, seizing thousands of women and children, and forcibly recruiting young men and boys into their ranks.
In a less publicised attack in November 2014, some 300 children were among about 500 people kidnapped from the town of Damasak, on the border with Niger, in the far north of Borno state.
Most are still missing.
The release of the 21 girls in October last year followed talks between Boko Haram and the Nigerian government brokered by the ICRC and the Swiss.
Three other girls have also been found. The first had a baby and was accompanied by a man she said was her husband but the military said was a Boko Haram suspect.
Shekau has previously said the girls would be released if militant fighters held in government custody were freed.
A total of 113 Chibok girls are now missing, although Shekau claimed last August that some had been killed in military air strikes.
On Friday, Britain and the United States issued a security alert warning of a Boko Haram plot to kidnap foreigners in the Banki area, which led to the suspension of aid flights to the town Saturday.