Kokkinakis saved four match points before finally converting on the sixth 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.
Skipper Lleyton Hewitt sprung a selection surprise, opting for Kokkinakis to lead Australia into battle on Thursday night ahead of the higher-ranked Alexei Popyrin and Jordan 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.
Kokkinakis's victory gave world No.9 Alex de Minaur the chance to send Australia into a third straight Davis Cup semi-final with victory over the fifth-ranked Taylor 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.
Thompson and Matt Ebden have been named to play Americans Rajeev Ram and Austin Krajicek in a potentially decisive doubles.
Ebden teamed with John Peers to defeat Ram and Krajicek in the Olympics Games final in Paris in August.
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.
Kokkinakis vindicated his captain's faith to steam through the opening set in 22 minutes, breaking Shelton three times.
