The Brisbane battler peeled off five straight games to take the first set and held an early break in the third, but couldn't maintain his level in a 2-6 6-3 6-3 loss on Saturday.
Majchrzak saved two break points at 3-3 to hold a crucial service game in the deciding set, before Duckworth leaked errors on his own serve in the next game to surrender the fatal break of serve in a tightly fought two-hour contest.
Earlier, Alex de Minaur took his frustrations out in a quickfire thumping to put Australia ahead 1-0.
Australia were upset in Ecuador earlier this year and must win this clash to progress to next year's qualifying round, where they can then book a place in the eight-team finals in Italy in November.
World No.9 de Minaur, 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 world No.231 Maks Kasnikowski.
The match lasted 56 minutes and 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.
Kasnikowski saved two set points on serve and forced de Minaur to serve out the first set, which he did with ease, before breaking instantly to set the tone for a one-sided second set.
"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."
Kasnikowski admits he was blown away in his first meeting with a member of the world's top 10.
"It was a unique experience, a really, really tough challenge and happy I put up a little bit of a fight in the first set," he said.
"In the second, Alex increased the level a lot and it was too much for me today.
"He started playing way faster and was making no mistakes."
Sunday's action will begin with a crucial doubles rubber, before de Minaur faces  Majchrzak in a repeat of the 's-Hertogenbosch final the Pole won in June.
Duckworth and Kasnikowski are then slated to play in the final rubber if required.
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