Debutant Corentin Moutet gave 10-times champions France the ideal start in Osijek by beating home favourite Dino Prizmic 6-4, 5-7, 6-1 before world number 57 Arthur Rinderknech topped big-serving 2014 US Open winner Marin Cilic 6-2, 6-4.
France will progress if they win any of the three possible rubbers on Saturday, beginning with the doubles.
In Groningen, hosts the Netherlands hope to emulate last year's run to the final, which they lost to Italy, but they already have a mountain to climb after losing both singles to Argentina.
Jesper De Jong was beaten 6-4, 6-4 by Tomas Etcheverry before world number 21 Francisco Cerundolo battled past Botic van de Zandschulp 7-6(1), 6-1 to put the visitors in charge.
Dutch captain Paul Haarhuis said his side will need to be perfect on Saturday.
"We can't afford any bad matches tomorrow, it's a deep hole we're in," he said. "But we still have a chance."
Austria are well-placed to spring a surprise in Hungary after also moving 2-0 ahead in Debrecen.
Jurij Rodionov, ranked 158th in the world, gave Austria the lead by stunning 56th-ranked Fabian Marozsan 6-2, 6-7(5), 7-5, and Lukas Neumayer then, also ranked outside the top 50, outlasted the experienced Marton Fucsovics 6-3, 3-6, 7-6(7).
Germany's Jan-Lennard Struff, who reached the fourth round of the US Open this month, put his side ahead against Japan in Tokyo with a hard-fought three-set win against Yoshihito Nishioka, and Yannick Hanfmann eased past Shintaro Mochizuki.
In Delray Beach, United States number one Taylor Fritz levelled the tie 1-1 against the Czech Republic after beating 17th ranked Jakub Mensik 6-4, 6-3.
"It's great. It's great obviously playing for Team USA," Fritz said. "The crowd tonight was great in Delray. I love being in Delray, and we needed this one, so it feels great to come out and deliver for the team."
Jiri Lehecka had given Czech Republic the lead after a dominant 6-3, 6-2 win over Frances Tiafoe.
Spain's clash with Denmark in Marbella begins on Saturday, as does Australia's home tie with Belgium.
Holders Italy are through to the Final 8 as hosts, with the seven winners from this weekend's qualifiers joining them.