Kazakhstan number one Kukushkin quickly disposed of Thanasi Kokkinakis 6-4 6-3 6-3 before Nedovyesov upset Nick Kyrgios 7-6 (5) 6-7 (2) 7-6 (5) 6-4.
Australia, bidding to reach the semi-finals for the first time in 2006, need to win Saturday's doubles just to stay alive heading into the concluding reverse singles.
The losses were compounded by the news that Bernard Tomic, who was banned for the tie after an outburst aimed at Tennis Australia officials, was arrested in the United States overnight.
Tomic was charged with trespassing and resisting arrest following complaints over a noisy penthouse party at a Miami Beach hotel, police said.
"Obviously that's the first thing that came up on my Facebook when I had a scroll," Kokkinakis said.
"We're just trying to do our own thing. We've got the tie, which I tried to focus on."
Kazakhstan reached the Davis Cup World Group for the first time in 2011 but have never made it past the semi-finals.
The winner of the best-of-five tie will advance to play either France or Britain in September.
(Writing by Julian Linden; editing by Sudipto Ganguly)
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