The feat, with New Zealand-based skiers Nick Begg and Tyrone Low, was only the second-ever attempt to ski down the face.
Senior Constable Brent Swanson of Lake Tekapo police said the men had been skiing along Aoraki-Mt Cook's summit ridge, near the lower peak, yesterday morning when one of them fell 600m vertically down the steep slope.
"One minute his partner was skiing behind him, and next minute he's looked back and he's gone," he said. "Basically he's slipped off and we don't know why."
Mr Swanson said skiing along the ridge was "right up there at the extreme level".
The survivor had "a bit of a scare" but was uninjured.
Mr Swanson said climbing conditions had been good and the weather was fine at the time.
A Department of Conservation alpine rescue team was dispatched after the survivor called emergency services about 8.30am yesterday. A rescuer, dangling on a strop from a Helicopter Line helicopter, picked him up shortly before midday. Mr Kastengren's body was recovered about half an hour later.
DoC Aoraki-Mt Cook alpine rescue team leader Jim Spencer, who has been an alpine ski guide for 15 years, said skiing on the mountain could be very dangerous and there was no room for error.
"If you were to slip, you would very quickly pick up a speed that you couldn't control."
Mr Spencer said the pair's accomplishment on the east face last week - which he probably would not attempt himself - was "very, very credible alpine skiing".
"Those boys were at the top of their game, I'd say."
Swedish honorary consul to New Zealand Frank Olsson said a police officer had named the man as Mr Kastengren. He said officials in Stockholm had been trying to get in touch with the family in Sweden, but he had not received confirmation his relatives had been reached yet.
Both Swedes have been involved in a number of ski-mountaineering firsts.
The mountain death is the second within a week after two climbers died on Mt Taranaki over Labour Weekend.
High-altitude deaths
There have been at least seven deaths on NZ mountains in the past three months:
August 12: Jamie Vinton-Boot, 30, died in an avalanche while climbing along the western face of The Remarkables. He was the 2012 NZ Mountaineer of the Year.
August 31: British-born Caroline Johnstone, 44, slipped on ice while retrieving her snowboard on Mt Ruapehu.
September 13: Duncan Rait, 36, fell 150m off a ridge near the Tasman Glacier at Aoraki-Mt Cook National Park.
September 14: Englishman Robert Buckley, 31, died after falling 700m near Mt Sefton Bivvy, at Aoraki-Mt Cook National Park.
October 28: Hiroki Ogawa, 31, and Nicole Sutton, 29, died after spending two nights in a snow trench after bad weather set in on Mt Taranaki.
November 3: Magnus Kastengren, 31, falls 600m to his death from Aoraki-Mt Cook while taking part in extreme skiing.