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Brothers Elias and Mikael Ymer helped Sweden sweep aside Canada, France fought back to down the Czech Republic and Croatia crushed Australia as the revamped Davis Cup opened on Thursday.
Holders Spain suffered a blow to their title defence before hitting a ball in anger with Carlos Alcaraz in quarantine after testing positive for Covid-19.
The rest of the Spanish team, who are missing Rafael Nadal, were given the all-clear after PCR tests to play Ecuador in Madrid on Friday.
Sweden were in imperious form in the Spanish capital against the Canadian 2019 finalists who went into battle without their top two players Felix Auger-Aliassime and Denis Shapovalov.
Elias Ymer, 25, got the show on the road, beating Steven Diez 6-4, 6-2 before his two-year younger sibling Mikael sped past Vasek Pospisil 6-4, 6-4.
"Today my forehand was blasting," said Elias. "I was hitting winners from everywhere!"
Robert Lindstedt and Andre Goransson then combined for a 7-6, 6-4 doubles win over Pospisil and Brayden Schnur to leave the Swedes needing a win over Kazakhstan on Saturday to progress to the quarter-finals for the first time since 2011.
In Innsbruck, where the surge in coronavirus cases and the subsequent lockdown in Austria meant that matches ware being played once again in an empty arena, France got off to a rocky start when 35-year-old veteran Richard Gasquet lost 7-6, 6-2 to Tomas Machac.
"Experience isn't everything. Form is the key and I wasn't up to scratch today," said Gasquet.
But France then came back as Adrian Mannarino beat Jiri Vesely 6-7 (1/7), 6-4, 6-2.
And Nicolas Mahut and Pierre-Hugues Herbert completed the comeback with a three-set defeat of Jiri Lehecka and Machac.
Great Britain complete Group C.
Australia, 28-time winners of the Davis Cup, have work to do after losing 3-0 to Croatia in Turin.
Borna Gojo beat Alexei Popyrin 7-6, 7-5 before Marin Cilic despatched Alex de Minaur 6-1, 5-7, 6-4.
Nikola Mektic and Mate Pavic then raced to a 6-3, 6-1 win over de Minaur and John Peers.
Hungary await their debut in this group D.
On Friday, as well as Spain v Ecuador, the United States open against Italy and Novak Djokovic's Serbia take on Austria.
The Davis Cup, which began life back in 1900 as a bilateral contest between Britain and the United States, has had numerous makeovers in the past, most recently in 2019 when it was reshaped to resemble a world championships.
This year's competition sees 12 qualifiers - Australia, Austria, Colombia, Croatia, Czech Republic, Ecuador, Germany, Hungary, Italy, Kazakhstan, Sweden and USA - joined by the previous edition's semi-finalists - Spain, Canada, Great Britain and the Russian Tennis Federation.
These are joined by two wild card nations France and Serbia.
These 18 nations are divided into six groups of three, with winners and the top pair of second places going into the quarter-finals pot.
The semi-finals and December 5 final will all be played in the Spanish capital, reward for the team that won the title two years ago after 2020 was cancelled due to the Covid-19 pandemic.