It took the tetchy Russian world number two 3hr 30min to subdue the unseeded American serve-volley specialist Cressy 6-2, 7-6 (7/4), 6-7 (4/7), 7-5 to set up a quarter-final against Canada's Felix Auger-Aliassime.
"It was long and it was not easy, the scoreline apart from the first set were all tough sets," said Medvedev, who clashed with the umpire and complained of his bad luck during the most severe examination of his title credentials so far.
"If I didn't win the fourth set I would have been in a tough mental shape," he admitted.
Earlier Cornet ended her last-16 hoodoo in her 63rd Slam and will now face Danielle Collins for a place in the women's singles semi-final.
The French veteran, who made her major debut in 2005 and turned 32 on Saturday, had reached the fourth round on five previous occasions, losing them all.
But she ended that run by upsetting former world number one Simona Halep of Romania 6-4, 3-6, 6-4 in energy-sapping heat on Rod Laver Arena.
"It feels amazing. That battle I had with Simona today and the heat," said Cornet, who knocked out world number three Garbine Muguruza in round two.
"After 30 minutes we were both dying and we kept going for two-and-a-half hours."
'Really tough'
Collins had an even longer battle, the American outlasting Elise Mertens of Belgium 4-6, 6-4, 6-4 across 2hr 51min of bruising action.
"It was really tough for me," said Collins, who made the semi-finals at Melbourne Park in 2019.
"I played a long match as well the other day so I had to make a lot of technical adjustments to be able to get myself moving around. That was a big challenge."
Aryna Sabalenka will round off the evening and stay on course for a first major if she can get past unseeded veteran Kaia Kanepi of Estonia.
The reward for the winner will be a quarter-final against seventh seed Iga Swiatek of Poland or Sorana Cirstea of Romania.
Men's ninth seed Auger-Aliassime overcame the 2018 Australian Open finalist Marin Cilic 2-6, 7-6 (9/7), 6-2, 7-6 (7/4) in 3hr 35min on John Cain Arena.
He joins Denis Shapovalov in the last eight to make it the first time two Canadian men have reached that stage at the Australian Open, and will now face Medvedev.
"It's amazing. It puts my belief even higher, I lost three times to Marin in the past, this is my first win against him and in an important moment like this," the Canadian said.
"I told him at the end of the match he's a champion, the way he handles himself and the way he plays and I knew he was going to come out with his best level and test me and make me earn my win.
"So I'm really relieved and happy to get through, especially the way I did it."
Later world number four Stefanos Tsitsipas, who is chasing a maiden major, plays American Taylor Fritz. The Greek is wary of the 20th seed.
"Big server, big hitter," said Tsitsipas. "He has weapons. I will really have to stay consistent and keep pushing him back."
Tsitsipas has weapons of his own -- he did not drop a service game against Benoit Paire in the third round.
"I feel like it's going to be really important to stay consistent with my shots and my serve," he added.
The other last-16 men's match on Monday features Australian hope Alex de Minaur against Italian 11th seed Jannik Sinner.
