It was the first joint space mission between the Soviet Union and the United States, carried out at the height of the Cold War, and was a prelude to the global co-operation seen aboard the current space station.
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.
His assistant, Ms Natalia Filimonova, told AFP that Mr Leonov died at Moscow's Burdenko hospital after a long illness.
Russian President Vladimir Putin sent his condolences to Leonov's loved ones, Kremlin spokesman Dmitry Peskov said.
British astronaut Tim Peake said Mr Leonov was "a hero to so many, pioneer of spacewalking and always full of wisdom and humour".
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.
Born in Siberia, the youngest of eight children, Leonov became a fighter pilot before being selected as one of the first potential cosmonauts. Apollo-Soyuz 19 was a prelude to the global cooperation seen aboard the current worldwide Space Station.
Two NASA astronauts are now outside the International Space Station on the second of several planned spacewalks to replace batteries affixed to the outside of the spacecraft.
"We had this tradition that he would give cosmonauts pep talks before they board the spacecraft", Kononenko said.
Astronauts replaced oversized batteries outside the International Space Station on Friday, as news broke of the death of the world's first spacewalker.
NASA on Friday offered sympathy to Leonov's family, saying it was saddened by his death.
Leonov, who will be buried on Tuesday at a military memorial cemetery outside Moscow, is survived by his wife, a daughter and two grandchildren.
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".