Novak Djokovic and Nikola Cacic lost a dramatic deciding tie-break in their doubles rubber as Serbia suffered a shock defeat by Germany in the Davis Cup Finals, while Daniil Medvedev helped the Russian Tennis Federation to an opening win.
World number one Djokovic had drawn Serbia level in their second group-stage tie by beating Jan-Lennard Struff after Filip Krajinovic's loss to Dominik Koepfer.
But the 20-time Grand Slam singles champion and Cacic slipped to a 7-6 (7/5), 3-6, 7-6 (7/5) defeat by Kevin Krawietz and Tim Puetz, who got over the line on their third match point in Innsbruck.
The matches in Austria are being played behind closed doors due to new coronavirus restrictions in the country.
Germany will qualify for the quarter-finals from Group F if they can beat Austria on Sunday, which would leave Serbia waiting to see if they progress as one of the two best runners-up.
Djokovic is hoping to cap a brilliant season, which saw him only miss out on a calendar Grand Slam with defeat by Medvedev in the US Open final, by helping his country win the Davis Cup for the first time since 2010.
The Russians, among the title favourites, made a perfect start to their campaign with a 3-0 victory over Ecuador.
World number five Andrey Rublev overcame a second-set blip against 291st-ranked Roberto Quiroz to win 6-3, 4-6, 6-1 in the opening rubber in Madrid.
Medvedev then ensured his team would win the tie by thrashing Emilio Gomez 6-0, 6-2 in under an hour.
Gomez was given a huge cheer by the crowd when he finally got on the board in the eighth game.
"I was ready for it," said the world number two. "I wanted to keep it clean from the first to the last point of the match.
"I'm really happy to bring a point to the team."
Rublev and Australian Open semi-finalist Aslan Karatsev completed the whitewash by taking the doubles rubber against Gonzalo Escobar and Diego Hidalgo 6-4, 4-6, 6-4.
Italy became the first team to reach the last eight, backing up their win over the United States by seeing off Colombia to wrap up top spot in Group E.
Lorenzo Sonego fought back to edge out Nicolas Mejia 6-7 (5/7), 6-4, 6-2, before Jannik Sinner brushed aside Daniel Elahi Galan 7-5, 6-0.
Earlier, British number one Cameron Norrie completed success against France in the second singles rubber, beating Arthur Rinderknech 6-2, 7-6 (10/8) in Britain's first tie.
Norrie, ranked 12 in the world, trailed 4-1 in the tie-break before rebounding to secure a match point at 6-5.
The 71st-ranked Rinderknech saved it but then let two set points slip before Norrie took the rubber on his second match point.
"For me, it's unbelievable to be back on the team playing for my country," said Norrie, who was playing just his third Davis Cup tie.
"It is also nice to prove myself again, to keep the pressure on and keep playing big matches."
Rinderknech then helped Mahut keep their slim hopes alive of securing one of the two best runner-up spots by outclassing Joe Salisbury and Neal Skupski 6-1, 6-4.
Dan Evans had earlier given Britain a great start by beating Adrian Mannarino 7-5, 6-4.
Kazakhstan could be surprise quarter-finalists, if they beat a much-weakened Canada team on Sunday, after pulling off a 2-1 win over Sweden.
The Swedish singles duo of brothers Elias and Mikael Ymer looked on course to win their second successive match after disposing of Canada 3-0 in the opener.
Elias eased past Mikhail Kukushkin in straight sets 6-3, 7-6 (7/4).
Younger brother Mikael was looking well poised at a set up and 4-4 in the second against the talented but unpredictable Alexander Bublik.
The world number 36, though, roused himself and produced some stunning tennis to reel off the next six games and take the rubber.
The Kazakhs completed their come-from-behind victory as Andrey Golubev and Aleksandr Nedovyesov breezed past Andre Goransson and Robert Lindstedt 6-3, 6-3.
Australia are all but out of the competition despite a battling 2-1 win against Hungary.