World No.2 Medvedev capped his amazing year with a 7-6 (9-7) 6-2 win over Marin Cilic in the second singles match to give Russia an insurmountable 2-0 lead over Croatia.
It was the fifth consecutive straight-set victory in the Davis Cup for Medvedev, who three months ago defeated Novak Djokovic to win the US Open for his first grand slam title.
"It was a pretty amazing two weeks... It feels (like) the best weeks of my career," Medvedev said.
World No.5 Andrey Rublev earlier had beaten Borna Gojo in straight sets to put the Russians ahead at the Madrid Arena.
Russia's other Davis Cup titles were in 2006 and 2002, and they are the fourth nation to win both Davis Cup and Billie Jean King Cup titles in the same year - after the women's team won in Prague last month.
The triumph capped a special week for Russia captain Shamil Tarpischev, the longest-serving Davis Cup captain.
He first captained in 1974, and appeared in his 100th match as captain in the semi-finals against Germany.
The Russian team is officially being called RTF (Russian Tennis Federation) in the competition amid its ongoing doping suspension in international sport.
After a tight first set, Medvedev broke serve twice in the second against the 30th-ranked Cilic to comfortably clinch the victory - and receive a standing ovation from the crowd.
"I loved the atmosphere. It was a really amazing two weeks because it's never easy to come here in the end of the season and compete. For sure it was the best two weeks of my career," Medvedev said.
Rublev had converted on his first match point to clinch a 6-4 7-6 (7-5) win over Gojo, who arrived at the Davis Cup Finals as the 279th-ranked player but had won all his three matches.
It was the third consecutive victory for the fifth-ranked Rublev after a loss against Spain's Feliciano Lopez in the group stage.
Gojo had no break opportunities throughout the match against Rublev, who converted one of his eight to grab the first set and eventually clinch the win in the second-set tiebreaker.
"It was super tough and I felt a lot of pressure, but it's good to put my country in the lead," Rublev told reporters.
Croatia got past Serbia and the top-ranked Novak Djokovic in the last four.
The Russians won a group that included defending champions Spain and Ecuador, then advanced past Sweden in the quarter-finals.
Croatia won a group with Australia and Hungary, then defeated Italy in the last eight.
The International Tennis Federation and the Kosmos Tennis group that is behind the competition said that next year four cities will host the group stage, with another venue - likely Abu Dhabi - hosting the knockout rounds.
with Reuters
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