Police say John Lowndes, a 59-year-old salesman from Stoke-on-Trent in England, and Kazuhiro Kotani, a 29-year-old from Hyogo, Japan, died while crossing Ball Pass.
The pair were roped together along with Erica Jane Beuzenberg, an acclaimed 41-year-old New Zealand climber and guide, Leask said. One of the climbers slipped and all three of them fell 200 meters (650 feet) to their deaths early Wednesday.
Bad weather in the mountains of New Zealand's South Island forced police to call off attempts to recover the bodies on Wednesday
"The bodies will remain where they are until the weather clears enough to remove them," said Christchurch police spokeswoman Maggie Leask. "This could be several days."
An unknown number of other climbers survived.
Mount Cook, which is 12,316 feet high, was first climbed in 1894 and attracts hundreds of climbers each year. The mountain can be treacherous. According to the New Zealand Press Association, 214 people have died on Mount Cook in the more than a century since the first climber reached its summit.
The Ball Pass is not considered one of the most difficult parts of the mountain.
"It is an unspeakable tragedy that such an experienced guide died on the job on what is perceived to be easy terrain," said Gottlieb Braun-Elwert, founder of the guide company that employed Beuzenberg.
Braun-Elwert said the climbers were descending a slope covered in snow and ice when one of them slipped and as he tried to brace himself against a fall, he unbalanced the guide and another climber.
They slid down the slope and off an ice cliff.
"The party then slid some further 200 meters (650 feet) down an icy slope before coming to a rest," he said. "All three were confirmed dead at the arrival of the search and rescue party."
Braun-Elwert said Beuzenberg was one of New Zealand's most experienced female climbers, who had worked as a guide for the company for 16 years.
In a separate climbing accident on North Island, also in New Zealand, a veteran climber's attempt to celebrate his 80th birthday on top of Mount Taranaki ended in tragedy when he fell to his death.
Police say the man, missing since Tuesday night and still to be formally identified, was found near the summit Wednesday.
The climber, who had been living in the United States but was born in Wales, is said to have had over 50 years of climbing experience in New Zealand.