Thanasi Kokkinakis held his nerve to beat Hungary's Zsombor Piros 6-4, 6-4 in the deciding qualifying singles rubber on Saturday to help Australia win the tie and seal a place in the Davis Cup Finals later this year.
Australia was level with Hungary before the final day after world number 30 Alex de Minaur's victory was cancelled out by Marton Fucsovics' defeat of Kokkinakis on Friday.
Hungary drew first blood on Saturday, after the doubles pair of Mate Valkusz and Fabian Marozsan eased past Luke Saville and John Peers, but De Minaur put in a solid performance to level the tie at 2-2, holding off Fucsovics for a 7-6 (4), 6-4 win.
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Kokkinakis entered the deciding match with a loss to his name in the tie, but was dominant from start to finish as he dispatched Piros in an hour and 24 minutes to earn the 28-times champions their third straight appearance in the Finals.
"I had full confidence in our guys," Kokkinakis said in his on-court interview. "De Minaur's our leader, an unbelievable week from him. He's put Australia on his back time and time again, so I had to do my part."
South Korea got off to a poor start against Austria on Friday, but was able to turn things around courtesy of a couple of impressive performances from Kwon Soon-woo.
Nam Ji-sung lost his singles rubber against Dennis Novak on Friday, before Kwon, who in September last year became the first South Korean to win on the ATP Tour since Lee Hyung-taik in 2003, managed to level with a win over Jurij Rodionov.
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On Saturday, the doubles pair Song Min-kyu and Nam won in straight sets against Lucas Miedler and Alexander Erler to give South Korea a 2-1 lead, with Kwon then beating Novak 7-5, 7-5 to seal the tie with a match to spare.
The 12 winning teams from the home-and-away qualifiers on Friday and Saturday will join Croatia, which secured automatic qualification for the 2022 Finals as last year's finalists, and wild cards Serbia and Britain.
Russia's title defence is uncertain after governing body International Tennis Federation (ITF) suspended the memberships of Russia and Belarus and withdrew their entries from team competitions following the invasion of Ukraine.