The world No.9 got Australia ahead 1-0 with a 56-minute masterclass against world No.231 Maks Kasnikowski, before Kamil Majchrzak edged James Duckworth in a tense two-hour affair.
Sunday's doubles will be pivotal before de Minaur and Majchrzak meet, just months after the Polish talent beat the Australian in a tight, three-set final in 's-Hertogenbosch.
If the tie is split 2-2, Duckworth and Kasnikowski are slated to play the decisive fifth rubber, with the winners progressing to next year's qualifying round and the losers eliminated altogether from the 2027 Cup race that culminates in Italy next November.
The hosts looked set to surge ahead 2-0 when Duckworth peeled off five straight games to take the first set and then break early in the third.
But Majchrzak broke straight back, then saved two more break points at 3-3, before a string of unforced errors cost Duckworth in his next service game.
"Kamil did a better job in key moments, a bit more aggressive and got on top of a couple more points and that was the difference," Duckworth said of the 2-6 6-3 6-3 loss.
"I didn't play a bad match, but there were a few moments that I'd like back."
Earlier, Alex de Minaur took his frustrations out in a quickfire thumping to shoot Australia clear.
The world No.9, keen to rediscover his mojo after a disappointing second-round US Open loss, lit up Pat Rafter Arena with a dominant 6-3 6-0 defeat of a wide-eyed Kasnikowski in his first date with a top-10 player.
De Minaur lost just six points in the second set as the Australian No.1 found range on his serve, angles from the baseline and got to the net to finish points with authority.
He roared after breaking for the first time in the fourth game and then, after a tight call went against him with his tail up at 4-1, steadied with back-to-back aces.
"That's what these ties are all about; get fired up, get the crowd involved and there's no better feeling," de Minaur said of his theatrics.
"We play outside of Australia for 10 months a year ... so being here, in front of a packed crowd, great atmosphere, it's one I've missed.
"The serve felt great ... I had some nice words to it and it's coming out to play."
Australian captain Hewitt lamented Duckworth's inability to hold serve after breaking to lead 2-1 in the third set.
"(Kamil) played better and better as the match went on," the former world No.1 said.
"It's everything to play for now. Since (Alex) got here he's got better and better. He was a class above today and it's going to be a big match, it doesn't matter what happens in the doubles.
"But Alex will be ready and he'll go out and play well again."
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