'Demon' drags hosts Australia into United Cup quarters
Also progressing into the last eight of the season-opening 18-team ATP-WTA event were Belgium and the Czech Republic, but Italy bowed out.
Hopes are high at home that De Minaur, fondly known as "Demon", can go deep at the Australian Open starting this month and he served warning with a 6-4, 6-1 win over the 18th-ranked Mensik.
That drew Australia level in the tie against the Czech Republic and De Minaur then paired up with Storm Hunter as they won the mixed doubles rubber in straight sets to progress.
The Czechs went through despite the 2-1 loss.
"I didn't start the year the way I wanted to," said De Minaur, who fell to Casper Ruud in his first group match.
"But something I’ve done really well in my career is bouncing back, so I’m happy I bounced back today."
In Perth, world number three Jasmine Paolini defeated France's Leolia Jeanjean 6-2, 6-3 for an opening win of the season, finally sealing the deal on her sixth match point.
But it was not enough as Italy joined France in exiting the competition.
Paolini, a two-time Grand Slam finalist, said: "First matches are always tricky.
"I'm happy how I felt on court today. We could not win the group this year but we did our best and fought to the last ball."
Wednesday's ties include quarter-finals between holders the United States and Greece, and Switzerland and Argentina, both in Perth.
World number two Iga Swiatek is in action in Sydney as Poland face the Netherlands.