Daniil Medvedev and Andrey Rublev secured a third title for the RTF following 7-6 (9-7) 6-2 and 6-4 7-6 (7-5) wins over Marin Cilic and Borna Gojo respectively.
It completed a double for the Russians after Liudmila Samsonova inspired the women's team to glory in the Billie Jean King Cup last month.
Appearing in their first Davis Cup showpiece since 2007, the RTF had lost only two matches during the finals – one of which coming during Saturday's win over Germany with the outcome already decided.
Rublev broke in game seven on the way to taking the opening set against Gojo 6-4.
The world number five was strong on his first serve, winning 36 of 39 points, and drew first blood for his nation after prevailing 9-7 in the second-set tie-break.
US Open champion Medvedev then took on Cilic, who was attempting to keep Croatia's hopes alive in what was their third final in five years.
The world number two's strong serve proved the difference during a tight opening set. Indeed, he hit seven aces on the way to edging his nose in front 9-7 on the tie-break.
Medvedev then broke the world number 30 in game four to take command of the second set.
And a crucial second break followed at 5-2 up to secure a hat-trick of Davis Cup crowns for his nation, who were also victorious in 2002 and 2006.