The Serbian team played without world No. 3 Novak Djokovic, who sat out Friday's singles match because of the flu. He is expected to play in Saturday's doubles match and Sunday's reverse singles.
Doubles specialist Nenad Zimonjic filled in for Djokovic against Youzhny, and took the first set before falling, 2-6, 6-3, 6-2, 6-4, on the hard court in Moscow.
Davydenko, ranked fourth in the world, put the Russians on the verge of eliminating Serbia with an up-and-down 6-1, 1-6, 6-3, 1-6, 6-2 win over Viktor Troicki.
Sunday's two singles matches include the marquee pairing of Davydenko against Djokovic while Youzhny is to play Troicki.
Russia finished second to the United States in the Davis Cup finals last year after winning the title in 2006.