Medvedev (N.2) brought his own the point of victory by controlling Marin Cilic (30th) 7-6 (8/6), 6-2.
Andrey Rublev (N.5) had previously given the advantage to the Russians by winning against Borna Gojo (279th) 6-4, 7-6 (7/5).
Russia's previous Davis Cup titles date from 2002 and 2006.
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