Two dozen rescue crews using helicopters and all-terrain vehicles were on the scene within 30 minutes of the crash.
Authorities said the man - a Swiss native who lived with his Canadian wife abroad - had returned with his family for a holiday.
Local police spokesman Florian Grossmann said: 'The cable car is not for passenger use.
"It is a freight vehicle that was being used on the day of the accident by an employee of the area.
"We are investigating how they came to be aboard and why."
An Alpine worker was arrested after he gave 'contradictory statements' to police at the scene, near the Waegitaler Lake.
It is understood he is the man who illegally allowed the family to use the freight lift.
The bodies of the parents were flown to a local morgue.
The child is recovering at a hospital where nurses and doctors have nicknamed her Eidelweiss - after the hardy mountain flower which only survives in the harshest of regions.
The police spokesman added: 'The child is out of danger and we are contacting relatives of the dead abroad.
"We have no cause as yet as to why the cable car halted and then suddenly tipped over.
"But it was not designed for passenger use, that much is certain."
- DailyMail