Get creative with layovers.
There are no direct flights between any city in North America and Sochi, Russia, which means you'll be stopping at least once – probably twice – on your journey to the Games. Aeroflot, the national carrier of Russia, flies direct from Los Angeles, Miami, New York, Toronto, and Washington, D.C. to Moscow's Sheremetyevo International Airport. Domestic carriers such as S7 Airlines and Transaero Airlines can take you the rest of the way to Sochi. Moscow has three major airports: Sheremetyevo, Domodedovo, and Vnukovo, so make sure you allow enough time if you must transfer between them.
You can also avoid Moscow altogether. Turkish Airlines flies direct to Istanbul from Boston, Chicago, Houston, Los Angeles, New York, and Washington, D.C. Istanbul is only 577 miles from Sochi and reachable via a one-hour flight on Aeroflot. Other creative places to connect include Kiev, Ukraine (via Ukraine International Airlines), and Chisinau, Moldova (via Air Moldova). To find the best possible deal, you should look at each segment of your flight independently and then piece together your own itinerary. That may mean flying from the United States to Istanbul on Turkish Airlines and from Istanbul to Sochi on Aeroflot. Your return flight may bring you back to Istanbul via Kiev on Ukraine International Airlines. It's going to be a hassle, regardless, so you might as well take advantage of the stopover. If that happens to be in Istanbul, you could visit the Strait of Bosporus, the place where Europe ends and Asia begins.