Grapes and wines represent the most important source of edible stilbenes and other phenolic metabolites, which demonstrate a wide range of valuable biological activities. However, there is no information about the profile and content of phenolic compounds in Russian wines. We firstly analyzed phenolics (stilbenes, phenolic acids, and flavonols) in some representatives of Russian wines, including eleven red and seven white Russian wines from Fanagoria, Krasnodarsky Territory. The Russian red wines contained six stilbenes (trans-resveratrol, cis-resveratrol, trans-, cis-piceid, trans-piceatannol, δ-viniferin), while the white wines contained only five stilbenes (cis-resveratrol, trans-, cis-piceid, trans-piceatannol, trans-resveratrol). More than a half of the total stilbenes in the wines (65% of all stilbenes) were presented by trans-piceid and cis-piceid, while trans-resveratrol reached 16% of all the stilbenes. The red wines also contained six phenolic acids and six flavonols, while the white wines contained six phenolic acids and only three flavonols. Myrecitin-3-O-glucoside, quercetin-3-O-glucoside, and myricetin were the major flavonols in the red wines, while dihydroquercetin-3-O-rhamnoside was the major flavonol in the white wines. The red wines contained markedly higher amounts of stilbenes, phenolic acids, and flavonols than the white wines. Thus, the data showed that young red Russian Fanagoria wines represent a rich source of phenolic compounds. The study also revealed that younger wines were more abundant in phenolics, and wine storage for six months in the dark at +10 °C led to a decrease in the total content of phenolics, primarily monomeric stilbenes and quercetin-3-O-glucoside and quercetin flavonols.