Spain's Nadal, 32, converted his sixth match point to win 6-1, 6-2, 6-4 in their third-round match in Melbourne, BBC Sport reported.
The 17-time Grand Slam champion dominated on serve, winning 82 percent of points on first serve.
De Minaur, seeded 27th, provided incredible resistance to save five match points in the final game, eventually succumbing to the sixth – after Nadal received a time violation from umpire Carlos Ramos – when he clubbed a forehand long.
"He's a big fighter and super quick. I played a great game until 40-0," Nadal said.
"I played a solid match and want to congratulate Alex for a great start to the season. I think he has a great future."
Nadal, who won the Australian Open in 2009, will face unseeded Czech Tomas Berdych next on Sunday.
Federer on fire
Meanwhile, Roger Federer delivered a serving masterclass to sweep aside American Taylor Fritz and reach the fourth round.
The defending champion won 6-2, 7-5, 6-2 under the roof on Rod Laver Arena at a rainy Melbourne Park.
The Swiss third seed, 37, is aiming to become the first man to win seven Australian Open titles.
Next up for him is Greek 14th seed Stefanos Tsitsipas, who beat Nikoloz Basilashvili 6-3, 3-6, 7-6 (9-7), 6-4.
"I wanted to get out the blocks quickly as I knew of the threat and his possibilities on the serve," said Federer, who was playing his 100th match on the Rod Laver Arena.
"The second set was tough. There were a few chances but he protected his serve well and it was close. We had some good points and it was fun. I wish him all the best for the future."
Defending champion Caroline Wozniacki was knocked out of the Australian Open as five-time Grand Slam winner Maria Sharapova provided a reminder of her ability with a three-set win.
Russia's Sharapova, seeded 30th, hit 37 winners in a 6-4, 4-6, 6-3 victory to reach the fourth round.
Wozniacki, 28, led 4-1 in the first set, only to see Sharapova battle back.
The third seed broke in the final game of the second to level, but Sharapova edged a tight third in Melbourne.
The 2008 Australian Open champion, 31, took her second match point to win in two hours and 24 minutes and set up a fourth-round meeting with Australia's Ashleigh Barty on Sunday.
"I haven't played many matches in the last year, especially against top players and these are the ones I train for, so it's really rewarding to win," said Sharapova, who served a 15-month ban for failing a drugs test.
Danish world number three Wozniacki, defending the ranking points won in Melbourne last year, is set to slide down the standings as a result of the defeat.
She was competing at a Grand Slam as the defending champion for the first time, but said the experience did not put on added pressure.
"I did not find it hard at all. I find it kind of pleasurable," she said.
"In tennis, you're one on one, you battle and you do your best. One day you win, some days you lose.
"I gave it everything I had today. She was just a little bit better than me and that was really it."
Sharapova has been unable to climb back into the upper echelons of the game since returning from a doping ban in April 2017, making just one Grand Slam quarterfinal at last year's French Open.
But against Wozniacki she proved she still has the talent – and, most importantly, desire – to cause problems for the world's best.
It was not Sharapova's biggest win in terms of ranking since returning, having also beaten Simona Halep – who was then second in the world – at the 2017 US Open.