I'm from Aust and flying to S.A. soon. I'm going from sydney to Chile. It stops over in NZ. Doesn't go through USA. This will prob suit you as you want to go through NZ anyway.
Check with Quantas, Aerolineas Argentinas, Lan Chile and Air New Zealand. 2 or 3 of them do have flights from South America to New Zealand/Australia.
But, the most interesting suggestion was:
It is possible and perhaps a little more interesting to fly from Chile to Easter Island (around $400.00) then onto New Zealand via Tahiti.
Lan Chile provides the cheapest flights to Easter Island. However you would have to chech to provider for onward flights.
Something Sarah and I will have to think about.