A never-say-die singles win from Alex de Minaur and a second doubles masterclass from Matt Ebden and Max Purcell ensured a 2-1 victory on Thursday to give the unbeaten Aussies maximum points from their two ties in Hamburg this week.
The resilient triumph after Jason Kubler's opening loss means that, even before they play hosts Germany in Sunday's final tie of the week, they look assured of a top-two place in their group which guarantees a quarter-final spot at the start of finals week in Malaga, Spain, in November.
"I'm really proud of the team," said Hewitt.
"We knew it wasn't going to be easy France had only the day before. Kubler left it all out there, then Alex never gave up and was able to find a way in the end, digging deep to give our doubles the opportunity to win it for us."
Hewitt's men had been staring at defeat as de Minaur found himself 4-2 and break point down in the final set to Benjamin Bonzi in the Rothenbaum Stadium.
With Kubler having lost the tie's opening rubber to French veteran Richard Gasquet 6-2 6-4, de Minaur had to pull out all the stops to rescue the situation with the French poised for an unassailable 2-0 lead.
The Sydneysider, straight out of the fighting mould of his team skipper Hewitt, did just that, scrapping for every point and then holding his nerve as Bonzi began to lose his.
De Minaur eventually prevailed 6-3 1-6 6-4 to leave it down to Australia's Wimbledon champions Ebden and Purcell to demonstrate their prowess again in the doubles decider with a 6-4 6-4 win over Arthur Rinderknech and Nicolas Mahut that never looked in doubt.
"Heart -that's what it is, that's what Davis Cup brings out of you," said de Minaur, as he reflected on how he got out of such a tight spot.
"Heart and passion - you'll do anything for your country. So I'm happy I got a win for my country."
As he kept failing to convert his break point opportunities early in the third set, de Minaur refused to get frustrated.
"I was trying to stay in the moment, give myself more chances. One of the big things I've been trying to work on is my mentality.
"I just kept telling myself I would give myself another chance and, hopefully, I'd take the next chance.
"I was able to do that late in the third set and I'm extremely proud of myself."
Kubler had earlier been unable to give Australia another flying Davis Cup start when he went down fighting against veteran Gasquet.
The 29-year-old debutant, who'd battled back to win his maiden Cup singles against Zizou Bergs in the 3-0 victory over Belgium on Tuesday, once again showed his spirit, saving five match points before losing to the inspired 36-year-old former top-tenner.
Kubler repelled the match points with some dazzling tennis, including a running forehand at full pelt and outlasting Gasquet in the longest and most thrilling point of the match.
But although he even earned a couple of break points of his own in that pivotal 10th and final game of the second set, Kubler could do nothing about an ace down the T on Gasquet's sixth match point.
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