In Valencia, the Czech Republic beat Spain 3-0 on Wednesday, with the hosts weakened when world No.2 Alcaraz decided to pull out of the squad to recover following his US Open semi-final defeat.
Tomas Machac cruised past Bernabe Zapata Miralles 6-4 6-4 and Jiri Lehecka then doubled their lead, downing Alejandro Davidovich Fokina 7-6 (7-5) 7-5 to put them in a powerful position in Group C alongside Novak Djokovic's Serbia, who won their tie on Tuesday.
The top two teams from each of the four groups will advance to the knockout rounds in Malaga, which take place in November.
Italy also endured a disappointing day in front of home fans in Bologna, as defending champions Canada secured a comfortable group A victory in a rematch of last year's semi-final.
World No.200 Alexis Galarneau beat Lorenzo Sonego 7-6 (10-8) 6-4 and the 158th-ranked Gabriel Diallo stunned Lorenzo Musetti 7-5 6-4 as Canada showed they can be a force even without their biggest names, Denis Shapovalov, Felix Auger-Aliassime and Milos Raonic.
Italy are without Sinner and Matteo Berrettini this week but still have three top-50 singles players in the squad.
In Split, the United States beat hosts Croatia 2-1. Mackenzie McDonald downed Dino Prizmic 6-4 6-2 before Borna Gojo levelled the Group D tie by beating world No.11 Frances Tiafoe 6-4 7-6 (8-6).
But the US pair of Austin Krajicek and Rajeev Ram edged out Ivan Dodig and Mate Pavic 7-6 (7-5) 6-7 (3-7) 6-2 in the decisive doubles rubber to ensure a 2-1 victory.
Hosts Britain defeated Australia 2-1 in Manchester, with the visitors relieved to see Matt Ebden and Max Purcell at least win the doubles after Jack Draper had defeated Thanasi Kokkinakis and Dan Evans beat Alex de Minaur, both in tight three-setters.
