They were from a group of 276 abducted in north-eastern Nigeria in 2014. Before the latest release, about 195 of the girls were still missing.
The number of Boko Haram suspects released by authorities remains unknown.
"I cannot express in a few words how happy I am to welcome our dear girls back to freedom," Mr Buhari told the girls in Abuja, according to his office.
"On behalf of all Nigerians, I will like to share my joy with you," he said.
Mr Buhari would have left earlier on Sunday to London but wanted to receive the schoolgirls, his spokesman Femi Adesina said.
Arriving in Abuja earlier, some of the girls looked tired and confused by all the attention after spending three years in captivity.
Before being taken to the capital, they were brought by road convoy from a remote area to a military base in Banki near the border with Cameroon.
Our reporter says that many families in Chibok will be rejoicing at this latest news, but more than 100 of the girls taken have yet to be returned.
"This is good news to us. We have been waiting for this day," Christian pastor Enoch Mark, whose two daughters were among those kidnapped, told Agence France-Presse.
"We hope the remaining girls will soon be released." It was unclear whether his daughters had been freed.
A statement from a spokesman for President Buhari earlier said he was deeply grateful to "security agencies, the military, the Government of Switzerland, the International Committee of the Red Cross, and local and international NGOs" for playing a role in the operation.
In a later BBC interview, presidential spokesman Garba Shehu added: "With all of these things together we negotiated over a period of several months, and at the end of it some of their [Boko Haram's] members were exchanged for the 82 girls."
After the Government Girls Secondary School in Chibok, Borno state, was raided in April 2014, more than 50 girls quickly escaped and Boko Haram then freed another 21 last October, after negotiations with the Red Cross.
The campaign for the return of the girls drew the support of then US First Lady Michelle Obama and many Hollywood stars.
Last month, President Buhari said the government remained "in constant touch through negotiations, through local intelligence to secure the release of the remaining girls and other abducted persons unharmed".
Many of the Chibok girls were Christian, but were encouraged to convert to Islam and to marry their kidnappers during their time in captivity.
Boko Haram has kidnapped thousands of other people during its eight-year insurgency aimed at creating an Islamic caliphate in north-eastern Nigeria.
More than 30,000 others have been killed, the government says, and hundreds of thousands have been forced to flee from their homes.