Police said human bones were spotted by a helicopter rescue pilot on the north face of the 14,690-foot Matterhorn in August 2013.
A police team recovered the remains and equipment, including an ice-axe.
They checked them against a database of some 280 climbers listed as missing in the region since 1926.
Thanks to DNA tests, experts were able to identify the remains as those of an Englishman who went missing in December 1979.
Police declined to reveal the 27-year-old man's name, citing Swiss confidentiality rules.
But he was identified by his family as Jonathan Conville, a former paratrooper who had gone on to work at a mountain sports centre near in the Scottish Highlands.
"Jonathan's remains were recovered, have been cremated and his ashes will rest in Zermatt," the Jonathan Conville Memorial Trust, set up by his family to encourage young climbers, said on its website.
British media reported that Conville was an instructor at an Outward Bound centre at Loch Eil in Inverness-shire, Scotland.
Conville was climbing with a Dutch friend when bad weather closed in on the pair as they were descending, according to reports.
The former parachute regiment soldier disappeared and his friend had to be rescued.
A Scottish newspaper, The Herald, reported that a Swiss forensic pathologist searched for the name Conville on the internet and made contact with relatives after discovering the Jonathan Conville Memorial Trust.
"As soon as I saw the email was from a Swiss pathology laboratory, I knew they'd found Jonathan," his sister, Melissa Conville, was quoted as saying by The Herald.
She and Conville's other sister Katrina Taee then travelled to Switzerland to identify their brother's equipment and provide DNA samples.
"It was poignant," Taee said.
"There was a mummified hand, with nails and skin, and cupped, as if it was waiting to be held," she told The Herald.
"It was 34 years on and I was holding my brother's hand," Taee said.
"It was bittersweet but wonderful.
"It took ages for the DNA results to come through but they confirmed what we already knew."
Technological advances have improved search operations and more cases of climbers reported missing are being solved.
In the case of Conville, a former Marlborough College student, his remains were taken for identification to a forensic laboratory at the University of Zurich.
The same lab is famous for analysing "Ötzi the Iceman", the oldest natural mummy in Europe, dating from 3,300 BC, found frozen in the Alps in 1991.
As Alpine glaciers melt due to global warming, the remains of long-lost climbers have increasingly emerged from the shrinking mountain ice.