But the Australians were belatedly indebted to a crucial doubles win for Matt Ebden and Max Purcell which reduced the final deficit to 2-1, leaving captain Lleyton Hewitt to believe his team, while really on the backfoot, can still make it to the Finals in November.
Cheered on by their home fans in Manchester on Wednesday (Thursday AEST), exciting new boy Jack Draper first outstayed a nervy Kokkinakis 6-7 (6-8) 6-3 7-6 (7-4) in a quality big-hitting duel before Dan Evans repelled a stirring comeback from Australian ace de Minaur to prevail 6-1 2-6 6-4.
But 2022 Wimbledon doubles champions Ebden and Purcell then rekindled their old partnership to beat Evans and Neal Skupski 7-6 (7-5) 6-4, a victory that at least left the team hopeful after a disappointing start to their bid to surpass last year when they lost to Canada after reaching their first final since 2003.
Needing a top-two finish in this week's round robin just to make it to the 'final 8' in Malaga again, the Aussies have to rebound on Thursday (Friday AEST) in a must-win duel with France and then also beat Switzerland on Saturday (Sunday AEST) if they're to be sure to make it.
But with every game, set and match result during the week counting, the 'M & M' team' knew their doubles victory could yet prove critical.
"We knew a win in this last rubber could be the difference at the end of the week so we really had to bring it and have to play great," said Ebden, fresh from his transatlantic crossing after playing in the US Open doubles final.
GB, in front of a partisan 9000 crowd in the AO Arena, raised eyebrows by dropping their No.1 Cameron Norrie and all-time great Andy Murray from their line-up but were left hailing their fresh prince, 21-year-old Draper after Kokkinakis had served for the match at 5-4 in the final set.
Draper also roared back from 4-2 down in the first match-deciding tiebreak he'd ever played at tour level as he finally prevailed after two hours 50 minutes.
Adelaide's Kokkinakis had been trusted by Hewitt to play the key starter role despite a disappointing display in last year's Finals but admitted: "I let my nerves get to me a little bit ... it's definitely a tough one. It stings for sure."
He'll surely now be benched for the France tie after a third straight Cup defeat with Hewitt admitting Kokkinakis had been a "little bit over-anxious". The captain conceded he had some thinking to do about his selection.
De Minaur, winner of seven of his last eight Davis Cup singles and playing his first match as a world top-12 player, missed early opportunities for a break against a then rampant, attacking Evans before his familiar never-say-die resilience hauled him back.
But Evans powered away brilliantly to a 4-0 lead in the decider and managed to hold off the Sydneysider after he'd forged back to 4-3, with Hewitt conceding de Minaur might "possibly" have been a bit fatigued after his recent heavy schedule.
"I'll do my best to get the boys up and hopefully we can turn things round," said de Minaur, who sounded somewhat glum.
He's odds-on to be leading the team again against France, who beat Switzerland 3-0 in their opener, probably facing southpaw Ugo Humbert, who beat him in Atlanta in July.
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