De Minaur's 6-4 6-1 drubbing of the youngest player in the world's top 20 levelled Australia's do-or-die clash with Czechia in Sydney at one rubber apiece.
Maya Joint earlier lost her women's singles encounter 6-4 6-1 to dual grand slam champion Barbora Krejcikova to leave Czechia one win away from a place in the quarter-finals.
But now the tie will go down to Tuesday night's mixed doubles rubber, which Australia must win to advance to the final eight of the $17 million teams' showpiece.
Answering a call from Australian captain Lleyton Hewitt, de Minaur will replace John-Patrick Smith and team with Storn Hunter for the deciding doubles match against Dalibor Svrcina and Miram Skoch.
While Joint disappointed, de Minaur dazzled in rebounding from his Ruud loss on Saturday night and improving his perfect record over the 18th-ranked Mensik to five from five.
"Obviously I didn't start the year the way I wanted to but I tried to not take that match too much into account," de Minaur said.
"I thought Casper played really well but something that I've done really well in my career is bouncing back, so I'm very happy with the way I bounced back today."
De Minaur's triumph also vaulted the 26-year-old to a career-high No.5 in the live ATP rankings.
If the world No.6 maintains that position when the new rankings are released on Monday, de Minaur will enter the Australian Open seeded fifth, behind only heavyweights Carlos Alcaraz, two-time defending AO champ Jannik Sinner, Alexander Zverev and 10-time winner Novak Djokovic.
On track to gain a first-time grand slam seeding at this month's Australian Open, Joint soared to No.32 in the world in a spectacular two-title season in 2025.
But facing Krejcikova - the 2021 French Open winner and 2024 Wimbledon champion - proved too tall of an order for the underdone 19-year-old in her first match of the year.
Joint had to sit out Australia's 2-1 weekend victory over Norway with illness, leaving Hunter to step up and prove the hero for the tournament hosts with victories in both the singles and mixed matches after de Minaur lost to Casper Ruud.
The accomplished Krejcikova, who also boasts a career golden slam in doubles, broke Joint in the opening games of both sets to have the home favourite on the back foot throughout, and wrapped up the match in one hour, 15 minutes.
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