The Argentines have reached the quarter-finals in Malaga, Spain, in November for the first time in five years.
They had to battle in Group D in Manchester after an opening loss to 2022 champion Canada. They beat their British hosts on Friday and overcame a gritty Finland, semi-finalists last year but without their best player.
Tomas Martin Etcheverry beat 703rd-ranked Eero Vasa 7-6 (7-5) 6-3, then Francisco Cerundolo outlasted Otto Virtanen 6-7 (4-7) 6-1 6-0 on Saturday. Argentina needed the doubles victory, and Maximo Gonzalez and Andres Molteni delivered against Patrick Kaukovalta and Harri Heliovaara 6-7 (3-7) 6-4 6-3.
The result means Britain have to beat Canada 3-0 on Sunday to advance at the North Americans' expense.
In Bologna, Brazil beat Belgium 2-1 after losses to hosts Italy and the Netherlands in Group A.
Three-set wins by Joao Fonseca and Thiago Monteiro followed by a doubles loss took more than six hours to complete on the indoor hard-court.
Brazil have a shot at advancing; the Dutch must lose to the unbeaten defending champions Italy 3-0 on Sunday.
In Zhuhai, China, the United States earned a seeding in the Final by beating Germany 2-1 and topping Group C unbeaten.
Both teams were already qualified for the Final but the 32-time winners grabbed a seeding awarded to the group winners.
Reilly Opelka saved two match points to beat Henri Squire 6-7 (4-7) 7-6 (11-9) 6-3 and Brandon Nakashima eased past Maximilian Marterer 6-4 6-2. The Americans dropped their first point in nine matches when Austin Krajicek and Rajeev Ram lost to Kevin Krawietz and Tim Puetz 6-1 7-6 (7-4).
"We have a good chance against any team there (in Malaga)," Nakashima said.
"There's a bunch of talented Americans playing really well right now."
In Group B in Spain, a tie between teams already out of contention saw 10-time champions France beat three-time champions the Czech Republic 2-1.
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