Cup debutant James Duckworth suffered a 7-6 (8-6) 6-4 loss to lowly ranked Roman Safiullan before world No.2 and reigning US Open champion Daniil Medvedev sealed the tie for Russia with a 6-4 6-2 win over Australian No.1 Alex de Minaur.
Tuesday night's defeat leaves Australia needing a miracle to reach the last four.
Russia improved to 2-0 after beating France in their competition opener, while Australia are 1-1 having upset Italy on Sunday.
Russia only need one more match win to confirm their progression to the semi-finals, meaning victory for Medvedev and Safiullan in the doubles rubber over Australia's nominated pairing of Luke Saville and John Peers will suffice.
Should Australia salvage a doubles victory, Lleyton Hewitt's team would then need to sweep France 3-0 on Thursday and hope Russia suffer a heavy loss to Italy.
Duckworth's straight-sets loss to the 167th-ranked Safiullan was particularly deflating for the tournament hosts.
Coming off his best season yet in 2021, Duckworth was unable to reprise the tennis that has propelled the 29-year-old into the top 50 for the first time.
Duckworth squandered a big first-set lead before falling 7-6 (8-6) 6-4 in the opening singles rubber at Sydney's Bank Qudos Arena as the world No.49's first serve deserted him under pressure.
De Minaur always faced an uphill battle repeating his weekend heroics against Italy's Wimbledon runner-up Matteo Berrettini.
Seemingly stung by his shock three-set loss to Ugo Humbert in his own season-opener, Medvedev played brilliantly to see off de Minaur in 79 minutes.
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