The 10-time champions took the lead on Tuesday when Adrian Mannarino battled past Dominic Stricker 3-6 6-1 6-4. Ugo Humbert sealed a comfortable 6-4 6-4 win over veteran Stan Wawrinka, who made his return to Davis Cup duty earlier in 2023 after seven years away.
"It's always difficult to play against Stan because he's a fantastic champion," Humbert said. "It's better to go on to the court 1-0 up. Adrian played a great match."
Nicolas Mahut and Edouard Roger-Vasselin then teamed up to beat Wawrinka and Marc-Andrea Huesler 6-2 6-2 in the doubles rubber.
France next face 2022 runners-up Australia on Thursday while Switzerland will aim to get their Group B campaign back on track when they meet hosts Britain the following day.
The top two teams from each of the four groups will advance to the knockout rounds in Malaga, which take place in November.
Djokovic will join the team in Valencia shortly for their last two Group C ties against Spain and the Czech Republic later this week, having claimed the US Open crown on Sunday to equal Margaret Court's record haul of 24 major singles titles.
The 36-year-old's teammates ensured there were no hiccups in their opener as Dusan Lajovic downed Hong Seong-chan 6-4 7-6 (7-3) and Laslo Djere battled past Kwon Soon-woo 4-6 6-2 6-2 to give Serbia an unassailable 2-0 lead in the tie.
Leo Borg, the son of 11-time Grand Slam winner Bjorn Borg, fought hard but was unable to give Sweden the start they wanted in their Group A tie against Chile as Cristian Garin prevailed 7-6 (8-6) 3-6 7-5 in Bologna.
Nicolas Jarry then settled the contest by defeating Elias Ymer 6-2 6-4 in 69 minutes.
In Group D, where ties are taking place in the Croatian city of Split, the Netherlands beat Finland 2-1 as Wesley Koolhof and Matwe Middelkoop edged out Harri Heliovaara and Patrik Niklas-Salminen 6-4 6-7 (5-7) 6-3 in the deciding clash.