De Minaur, who is looking to become the first Australian man to claim the Norman Brookes Challenge Cup since Mark Edmondson's triumph 50 years ago, will face top seed Carlos Alcaraz and needs a maiden win over the Spaniard for a place in the semi-finals.
"The last two matches I have hit the ball extremely well," said De Minaur.
"So I'm super pleased with my level, I'm excited for the next one. That's going to be a big one, right? I'm going to have to come out all guns blazing and I'm excited for a battle against Carlitos."
The first meeting between two top-10 men's players at the Melbourne Park major this year pitted Bublik's unpredictability against De Minaur's court coverage, and it was the Australian who landed the first blow by winning a tight opening set.
De Minaur hammered a forehand winner that sent Bublik the wrong way to earn two set points, and the 26-year-old benefited from a double fault from his Kazakh opponent at the most vital moment to grab the set at Rod Laver Arena.
"I knew what I was in for today," De Minaur said.
"Bublik is a hell of a player, he has so much firepower. I lost to him the last couple of times so I made sure I was ready to go from the first point to the last.
"I was locked in and it was all about neutralising his big groundstrokes and doing my best to get him on the move. It all worked perfectly so I'm super happy."
With the bit between his teeth and Bublik's level dropping a few notches, De Minaur raced to a 4-1 lead in the second set and pounced on his first set point to put one foot in the quarter-finals to the delight of the home fans.
The unorthodox Bublik dipped into his bag of tricks in the third set - the 28-year-old was unsuccessful with his 'tweener' but more fortunate with an underarm serve - but De Minaur did not wilt.
"I'm going to do my best to sleep in, get a good coffee to start the day," De Minaur said about his plans on Monday.
"As far as preparation, I've hit enough balls recently so I'm not going to need to do much. A nice easy session, relax a little bit, enjoy my time in Melbourne and get ready for an absolute battle in a couple of days' time."