Air France and KLM one-stop flights LAX to Moscow transiting via Amsterdam or Paris are $590 round trip. These low fares are available Sunday through Thursday from January into early May 2016. Friday and Sunday travel is about $40 more. It is possible to schedule a long layover in Amsterdam outbound, inbound or even both. There are also overnight layover itineraries for Paris.
Aeroflot is also a Skyteam Alliance airline and flies Los Angeles to Moscow nonstop. Their low price is around $688.
KLM Los Angeles LAX – Amsterdam AMS – Moscow SVO
$590.12 Sunday April 3 – Sunday April 10
Earn 4,124 Flying Blue miles in economy V class on this $590.12 fare.
This itinerary has a 9-hour layover in Amsterdam on the outbound and a 22:45 hour layover in Amsterdam on the return. Other flight options reduce the outbound flight layover to a couple hours with better arrival time in Moscow. The 23-hour layover in Amsterdam on the return is the kind of itinerary I love for a free city layover on a low cost fare.