There is wide availability across several airlines for under $800 flights to Johannesburg JNB South Africa and several other airports around $900 round trip. Oneworld Alliance member Qatar and SkyTeam Alliance Saudia are the low fares around $787 for LAX departures from September all the way to next March 2017. Saudia increases its LAX flights on June 1, 2016 from 3 flights to 5 flights per week.
Etihad and Emirates are also in the low fare mix around $800.
Other US and European major alliance airlines have jumped in with Delta, Air France, KLM, Swiss, Lufthansa, British Airways, and Virgin Atlantic offering fares around $900 round trip.
A year ago I stated a low airfare from California to South Africa was anytime it dropped under $1,000. That bar became a common fare in 2016 as long haul international travel dropped in price for regions of the world like Africa and Australia/New Zealand that rarely were under $1,000 in the past.
Etihad will fly you to Cape Town CPT for $885 round trip via LAX – Dublin, Ireland – Addis Ababa, Ethiopia where you have a long layover and time for a hotel bed before completing the last leg to Capetown. That 16 hours of flight time with a short one hour transit in Dublin, Ireland will be a rough. Preferable I think is planning for a stopover on one of the mideast carriers or a long layover in Europe to get horizontal in bed for a night before continuing on to South Africa.
Google Flights Los Angeles LAX to Johannesburg, South Africa JNB
November 9-24, 2016
Qatar Airways Los Angeles LAX – Doha, Qatar DOH – Johannesburg, South Africa JNB
$786.03 Wed October 5 – Wed October 19
A stopover in Doha is allowed in one direction for $100.
LAX-DOH-JNB-DOH-LAX = 24,338 flight miles in economy O class.
Qatar Economy O earns 25% miles with several Oneworld Alliance airlines including AAdvantage.