$586 - Flights to Moscow from San Francisco
The cheapest flights are with Delta Airlines, but both KLM and American offer cheap flight as well.
The American Airlines discount offer is for flights traveling between August 31st and September 30th. Flights into Russia must depart on Tuesdays or Thursdays, and returning flights must be on Mondays and Wednesdays.
With American, each way costs $301 not including taxes and fees, so Delta and KLM are still cheaper regardless.
Flight time between San Francisco and Moscow is typically between 20 and 24 hours including layovers.