Russia have not won the Davis Cup since 2006 and failed to reach a single quarter-final between 2011 and 2018.
The second-ranked Medvedev brushed aside Jan-Lennard Struff aside 6-4, 6-4 to seal the Russian victory after Rublev took the lead with a 6-4, 6-0 win over Dominik Koepfer.
Davis Cup: Daniil Medvedev leads the way as Russia defeat Sweden to enter second successive semi-final
Russia will on Saturday face Germany for a place in Sunday's final with Novak Djokovic's Serbia meeting neighbours Croatia in the other semi-final on Friday.
Recently crowned the season-ending world number one for a record seventh time, Djokovic just missed out on winning all four Grand Slams in a single year.
World No 2 Medvedev moves on to the round of 16 where he will face Bulgaria's Grigor Dimitrov, who cruised past American Reilly Opelka.
Norrie saved eight of nine break points he faced to reach his fifth ATP final of the year.
Andrey Rublev set up a showdown against Cameron Norrie of Britain, who scored his second straight upset win over a seeded player this week, defeating No 4 Denis Shapovalov
Next up for Rublev is a match against No 6 seed Diego Schwartzman who beat Lloyd Harris.
Laver Cup 2021: Alexander Zverev, Andrey Rublev give Team Europe title with three matches still left
There were still three singles matches remaining, but Team Europe already had eclipsed the 13 points needed to win the Laver Cup.
Andrey Rublev, Casper Ruud and Matteo Berrettini won their singles matches as Team Europe jumped to a 3-1 lead over Team World following the opening day of the fourth edition of the annual tennis showdown.
Day 5 of US Open had its share of fantastic matches and upsets with Tsitsipas, defending champion Osaka and Rublev exiting the tournament.
Fernandez, who won her first WTA title in March at Monterrey, had never beaten so high-ranked a rival as third-rated Osaka and the same was true for Alcaraz when he sent home the men's world number three.
Tsitsipas registers another victory, Muguruza and Medvedev move further. Check out the key moments from US Open 2021
Osaka seeking her fifth career Grand Slam title and third US Open crown in four years, plays a second-round match against 145th-ranked Serbian qualifier Olga Danilovic.
US Open 2021: Novak Djokovic motivated by 'unique opportunity' as challengers look to thwart attempt at history
Only two men have won all four majors in a calendar year. Don Budge was first in 1938 and legendary Rod Laver managed the feat twice – in 1962 and 1969. In the 52 years since, no man had won the first three Slams of the year until Djokovic has in 2021.
US Open 2021: Novak Djokovic faces potential Wimbledon final rematch against Matteo Berrettini in quarter-finals
Under a men's singles draw released Thursday by the US Tennis Association, the 34-year-old Serbian top seed was placed in a draw quarter that could see him meet Australian fourth seed Alex de Minaur in the fourth round.
Cincinnati Masters: Andrey Rublev ends losing streak against Daniil Medvedev, to face Alexander Zverev in final
World No 7 Rublev had never won so much as a set in four previous losses to Medvedev, top seed in the prestigious hardcourt tournament that serves as a tuneup for the US Open.
Cincinnati Masters: Medvedev books semis spot with win over Carreno Busta; Barty beats Krejcikova to reach last-four
Medvedev, last week's Toronto champion, won his eighth match in a row as he prepares for a final four contest against compatriot Andrey Rublev.
Medvedev finished with 10 aces as he reached his eighth quarter-final at the Masters level – all achieved since 2019.