Kokkinakis saved four match points before finally converting on the seventh of his own in a tension-filled and vital 6-1 4-6 7-6 (16-14) win in the opening singles rubber of Australia's quarter-final tie in Malaga on Thursday (Friday AEDT).
But, clearly hindered by his hip injury, brave Australian No.1 Alex de Minaur then succumbed 6-4 6-4 to Taylor Fritz in the second singles match as the world No.4 levelled the tie at one apiece.
American Taylor Fritz (R) with Australia's Alex de Minaur after levelling their Davis Cup tie. (AP PHOTO)
Jordan Thompson and Matt Ebden will play Americans Rajeev Ram and Austin Krajicek in the decisive doubles rubber.
Ebden teamed with John Peers to defeat Ram and Krajicek in the Olympic Games final in Paris in August.
Lleyton Hewitt had sprung a skipper's selection surprise, opting for Kokkinakis to lead Australia into battle ahead of the higher-ranked Alexei Popyrin and Thompson.
With title glory in Montreal in August, Popyrin was the first Australian since Hewitt in 2003 to win a Masters 1000 crown, before taking down the so-called GOAT Novak Djokovic at the US Open four weeks later.
Thompson has also cracked the world's top 30 for the first time with his own breakout season.
But, boom or bust, Hewitt continued to back Kokkinakis, despite him being ranked more than 50 places below both Popyrin and Thompson.
Kokkinakis was fresh off winning the NSW Open's ATP Challenger Tour event in Sydney after also upsetting 2023 Australian Open runner-up Stefanos Tsitsipas at the US Open in September.
And the 28-year-old delivered to give Australia a 1-0 lead in the do-or-die quarter-final Cup tie.
Ben Shelton looks on in frustration after losing to Kokkinakis. (AP PHOTO)
Kokkinakis's victory gave de Minaur the chance to send Australia into a third straight Davis Cup semi-final with victory over Fritz in the second singles match.
Fritz edged de Minaur in the ATP Finals group stages last week in Turin to pull back their head-to-head record to 5-4 in the Australian's favour, before accepting it was a "nightmare" prospect facing the speed demon of tennis.
But the US Open runner-up repeated his victory from last week to send the Cup last-eight tie into a doubles shootout.
Chasing a first Davis Cup title since 2003, Australia are bidding to make a third consecutive final after losing to Canada and Italy in the past two deciders.
