Spaniard Nadal, the 2009 winner and number two seed, gained a routine 6-3, 6-4, 6-2 win over Italy's 16th seed Fabio Fognini at Melbourne Park, BBC Sport reported.
Nadal will face the Greek fifth seed Stefanos Tsitsipas in the last eight on Wednesday.
Nadal's victory means he has reached a Grand Slam quarterfinal for the 43rd time, going on to win 20 titles, although his only success in Melbourne was 12 years ago over Roger Federer.
He came into this tournament with a back injury, having withdrawn from the ATP Cup earlier this month, but has not dropped a set in his four matches.
"I played a good game when he broke me and was lucky to have the break back immediately and save a game from love-40 so I was a bit lucky," said 34-year-old Nadal.
"It was humid and warm but the conditions are quicker during the day and for me it's better to play in the day as I like the ball flying but I need to adapt for all the conditions."
Russians make history
Medvedev beat American world number 192 Mackenzie McDonald 6-4, 6-2, 6-3 to claim his 18th consecutive victory and it means there are three Russians in the quarterfinals of a Grand Slam for the first time in the Open era.
"I want more all the time, but step by step, so this is amazing," said Medvedev, who only needed one hour and 29 minutes to defeat McDonald on Margaret Court Arena.
Medvedev will face compatriot Andrey Rublev in the last eight. Seventh seed Rublev was leading 6-2, 7-6 (7-3) when Norwegian opponent Casper Ruud retired with an injury.
Aslan Karatasev, who had to win three matches in the qualifying event in Doha, won on Sunday as his dream Grand Slam debut continued.
No Russian male has won a Grand Slam since Marat Safin in Melbourne in 2005.
World number one Ashleigh Barty only needed 71 minutes to defeat American Shelby Rogers and move into the women’s quarterfinals.
Barty, who is aiming to become the first Australian to win the women's singles since 1978, triumphed 6-3, 6-4.
There has not been an Australian winner of the women's singles since Christine O'Neil beat American Betsy Nagelsen 43 years ago.
Barty, 24, won the French Open in 2019 and reached the semifinals in Melbourne last year before losing to eventual winner Sofia Kenin.
As a result of the coronavirus restrictions, Barty opted not to play in the behind-closed-doors US Open in August and September and then also chose to not defend her title at Roland Garros.
But, playing in her first Grand Slam in a year, she has not dropped a set in any of her four matches in Melbourne.
She will face 25th seed Karolina Muchova, after the Czech Republic player knocked out Belgium's 18th seed Elise Mertens, winning 7-6 (7-5), 7-5.
"It's really exciting to be through to another quarterfinal," said Barty.
"If you had told me this would have happened in preseason I would've had a massive smile on my face.
"I just love being at home, I love being in Australia and sharing it with my friends and family."