Mr Destro added that the girl initially answered questions when officers rushed to her aid but "did not make it".
Police have launched an investigation and are due to question the teenager's classmates who boarded the lift with her.
The school's principal, Rajinder Singh Sandhu, broke the news of the teenager's death to her family. In a statement, he said the half-term excursion was led by the vice principal of the school with the help of two assistant head teachers and "a massively experienced team of eight staff". He added that the school is "doing everything possible to support the family in any way possible".
A Foreign & Commonwealth Office spokeswoman said: "We can confirm the death of a British National on Friday 22 February in Italy. We are providing consular assistance at this sad time."
Claviere is a ski resort in the province of Turin which crosses the French border.