The flight and spacewalk were successful on the whole, but his return to Earth almost ended in tragedy when Mr Leonov and another cosmonaut, Mr Pavel Belyayev, crash-landed in the Siberian woods.
"This is a loss for the whole planet", said Russian cosmonaut Oleg Kononenko.
Leonov flew a second time in 1975, when he was the commander of the Soviet half of the landmark Apollo-Soyuz test project, the first Russian-American joint space mission.
Tass reports he died after battling a long illness.
In 1975, Leonov took part in the historic Soyuz-Apollo spaceflight which, put an end to the "space race" between the USA and Soviet Union.
Russian President Vladimir Putin sent his condolences to Leonov's widow and daughter, calling him a "true trailblazer" and a "heroic person".
"Putin always admired Leonov's courage and thought he was an extraordinary man", Peskov told Russian news agencies.
On Twitter, Nasa added: "His venture into the vacuum of space began the history of extra-vehicular activity that makes today's Space Station maintenance possible".
Leonov was born in 1934 in a large peasant family in western Siberia.
Since Leonov's feat, there have been 227 spacewalkers including 14 women.
As U.S. astronauts Andrew Morgan and Christina Koch wrapped up a successful seven-hour spacewalk, the rest of the station crew paid tribute to Leonov. He was such a legend that the late science fiction writer Arthur C. Clarke named a Soviet spaceship after him in his 2010 sequel to 2001: A Space Odyssey.
Leonov was a close friend of Yury Gagarin, who became the first human to go to outer space in 1961.
Leonov helped train other cosmonauts before retiring in 1992, a year after the Soviet Union collapsed.
Alexei Leonov was known as Cosmonaut No 11.
"It's a huge tragedy for us all", said the wife of fellow cosmonaut Boris Volynov, Tamara Volynova.
He helped many cosmonauts and was a "great storyteller", space historian Alexander Glushko said.
"One of the first cosmonauts of the world space era, Alexei Leonov was committed to his Motherland and his cause, his name is lettered in gold in the world space exploration history", Roscosmos said.