Via koika on Flyertalk you can fly from Sydney to Los Angeles for under US$1000 roundtrip on Air New Zealand (connecting in Auckland).
Brisbane departures work as well (and One Mile at a Time notes that Melbourne works), but you do need to originate in Australia. Roundtrips starting in the U.S. don’t price this way.
If you don’t want Los Angeles as a destination it should price as an add-on with another city as the destination.
Air New Zealand in Sydney
This fare will not come up on the Air New Zealand website but it’s bookable on Expedia. It books into Z and is broadly available even during peak periods. Air New Zealand is a Star Alliance member so you can credit miles to United or other Star partners, and they have partnerships outside the alliance as well.
This certainly won’t last long at all. Just in case the airline decides they want to renege on such a deep discount offer, a case of seller’s remorse, I’d wait awhile after buying tickets before making additional non-refundable travel plans around the fare.
Update: it happens that Air New Zealand launched a fare sale for their Australia routes today, though I don’t think this deal will last as long as the planned sale.