World-renowned mountaineer Alex Lowe had been climbing the 8,013m (26,290ft) Shishapangma peak in Tibet in October 1999 with cameraman David Bridges when they were hit.
They were found by two climbers last week, still trapped in the ice.
Mr Lowe's wife, Jennifer Lowe-Anker, said they had been "frozen in time".
Mr Lowe, 40, was considered one of the greatest mountain climbers of his generation and was well known in the climbing community for having rescued several other climbers.
He and Mr Bridges, 29, were close friends as well as climbing partners, and had been scouting a route up Shishapangma, the 14th highest mountain in the world.
The discovery of their bodies was announced on Friday by Mrs Lowe-Anker, who now runs a charitable foundation in her late husband's name with her husband, Conrad Anker.
Mr Anker had been on Shishapangma with the two climbers on the day of the avalanche but escaped with minor injuries. He and other climbers spent days searching for the missing pair.
He married Alex Lowe's widow in 2001 and adopted her three sons with the mountaineer.
The couple were in Nepal with their charity last week when they received a call from climbers David Goettler and Ueli Steck, saying they had found two bodies "still encased in blue ice but beginning to emerge from the glacier", said Mrs Lowe-Anker.
They described the clothing and equipment to Mr Anker, who said it was "undoubtedly" the missing pair, said the charity's statement.
Mr Anker told Outside magazine that it was "fitting" the men were found by climbers.
"It wasn't a yak herder. It wasn't a trekker. David and Ueli are both cut from the same cloth as Alex and me."
The bodies are now expected to be recovered from the mountain.
Mr Anker said the discovery "brings closure and relief for me and Jenni and for our family".
Mrs Lowe-Anker said she and her husband and sons with Mr Lowe would now travel to Shishapangma, saying: "It is to put Alex to rest."
The bodies of two American climbers have been found in a glacier in the Himalayas, 16 years after they were killed by a huge avalanche.