Mexican officials initially said the pair were US diplomats, but an American source referred to them as embassy employees.
One of the climbers suffered a fall, forcing the other to seek help from the US embassy, according to the civil protection authority for the state of Puebla.
The rescue operation began on Sunday, but had to be called off because gusts of wind made it too dangerous for helicopters.
A helicopter finally managed on Monday to reach 26-year-old Nathan Cahill, who was airlifted to a Mexico City hospital, Mexican officials said.
Rescue workers recovered the other climber's body on foot.
"The strong gusts of wind and the area where (the body) was recovered, close to the summit, made it too difficult by helicopter," a civil protection authority official said.
Best known as Pico de Orizaba, the 5 610-meter mountain on the border between the states of Puebla and Veracruz attracts professional climbers from around the world as well as poorly equipped amateurs.
In November, authorities recovered the body of an American mountaineer from the same summit, following a risky rescue operation that injured one worker.
The mountain gained notoriety in 2015 following the discovery of two unidentified mummified bodies at an elevation of 5 200m - that have not been recovered due to their precarious location - and a third now housed in a local museum.