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The Russian Tennis Federation won their third Davis Cup title with a victory over Marin Cilic.
Rublev had defeated Gojo in the first singles match to put the Russians ahead.
He won his maiden Grand Slam title at the US Open, and was perfect throughout the tournament.
The Russians are the fourth nation to win both the Davis Cup and the women's equivalent team competition in the same year.
Cilic, who was part of the winning Croatian team in the US Open, was the first to test Medvedev.
The Croat built up a 4-2 lead in the first-set tie-break, but his forehand lost its way in the closing stages as the other player settled his nerves.
Cilic's serve was off in the early part of the second set, with Medvedev breaking him in the first game.
The 25-year-old said in his on-court interview that he was more happy for the team than himself.
It was the best two weeks of my career when I came at the end of the season.
At the start of the season, he was part of the team that won the ATP Cup.
