The government freed five top Boko Haram commanders in the deal, said Sen. Shehu Sani, who was part of the negotiating effort.
"There's no price too high to pay," Sani said.
The released girls were received at the airport on behalf of President Muhammadu Buhari by Alhaji Abba Kyari, his chief of staff.
"Welcome our girls, welcome our sisters," Kyari said to the girls, who sat quietly in chairs. "We are very glad that you are back, and every Nigerian today must be forgetting every other hardship and suffering, because this is a very joyous moment. Welcome, welcome, welcome."
They generally appeared to be in good physical condition. One has a broken arm and another has a leg injury, Nigeria's information minister, Lai Mohammed, told CNN.
More than 100 girls remain in Boko Haram custody and negotiations with the terror group continue, the government said.
276 girls were kidnapped three years ago
The girls, ages 16 to 18, are believed to be among 276 forced from their beds by Boko Haram militants in the middle of the night in April 2014. The kidnapping from a boarding school in the town of Chibok sparked global outrage and the social media movement #BringBackOurGirls.
"I am very, very excited with this development. I cannot even sleep tonight," said Yana Galang, whose daughter, Rifkatu, was among the girls kidnapped.
Galang said they don't yet know who has been released, "but we're very happy that many have been freed."
"I hope and pray that my daughter is among these released girls," she added.
President to meet with girls
Buhari delayed a previously scheduled trip to London for medical treatment to meet with the girls in Abuja on Sunday, his office said.
"The President has repeatedly expressed his total commitment towards ensuring the safe return of the #ChibokGirls, and all other Boko Haram captives," Buhari's office said in a statement.
"This evening I received 82 of our daughters, who have just regained their freedom after three years in Boko Haram captivity," Buhari said in a tweet.
Sani told CNN the negotiating team included Swiss intermediaries, Nigerian security forces and a mediator.
He said the negotiations came in three parts.
The first was the release of 21 girls last October
, which he said was designed to "build confidence on both sides."
Discussions to free the remaining 113 girls should begin at the end of the week.
Previously, as many as 57 girls escaped almost immediately after the mass abduction. Another girl was found in May 2016 when she wandered out of a Nigerian forest
asking for help, according to witnesses.
Red Cross acts as 'middle man'
On Sunday, The International Committee of the Red Cross tweeted, "Acting as a neutral intermediary, we @ICRC transported 82 #ChibokGirls to the government of #Nigeria" and included a photo of six Red Cross SUVs on a dirt road.
The Red Cross also tweeted a photo of some of the girls walking onto a helicopter departing Borno State.
"(Given) our long historical independent neutral stance, it's common for us to be the middle man in hostage releases," said Jason Straziuso, spokesman for ICRC for East Africa. "We provide transportation from Boko Haram to the Nigerian government."
"We will continue to meet with the girls and monitor their health," Straziuso said. "Our concern for the girls' well-being will continue in the coming months."