Gonzalo Escobar and Diego Hidalgo of Ecuador outperformed Ben Mclachlan and Yasutaka Uchiyama of Japan 7-6 (7-3), 6-3 in a deciding doubles match in Miki, western Japan.
Ecuador, ranked 27, who took a 2-0 lead on Friday against world number 17 Japan, added another victory to win their qualifier 3-0 and advance to the Madrid finals in November.
The remaining two singles were cancelled on account of Ecuador's three wins.
Japan were without two star players — injury-hit Kei Nishikori and world number 48 Yoshihito Nishioka, who decided not to join the team to avoid the possibility of being quarantined due to the outbreak on his return to the US.
The Davis Cup matches were held without spectators as a growing number of sporting events across the world are affected by the outbreak.
Sounds of smashing the ball echoed in the empty arena as ball kids wore gloves and used baskets for players to toss their towels in so they would not have to touch them.
A limited number of people allowed to enter the arena, including journalists, were required to check their body temperature, with some organisers wearing masks.
The spread of the novel coronavirus has led to the postponement of sporting events in Japan, raising fears that the outbreak could affect the Olympics due to begin on July 24 but Tokyo 2020 organisers said that preparations were "continuing as planned".
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