Rublev gave up a miserly nine games in his straight-sets win, the Russian delivering a ruthless 6-4 6-2 6-3 victory that left world No.30 Evans scratching for answers.
He will next face ninth-seeded Dane Holger Rune, who overcame a nasty fall early in the second set to eliminate Frenchman Ugo Humbert 6-4 6-2 7-6 (7-5) in their Saturday clash.
Mid-match Evans showed his class when he gave 25-year-old Rublev one of his bananas but he may have regretted such generosity.
"I asked the ball boy for a banana but Danny asked for bananas earlier and he had two so he threw me one," a smiling Rublev said.
"He helped me with some energy for sure - I win an extra two games because I eat the banana."
Balancing touch and finesse with some power hitting, Rublev hit 60 winners to 20 from Evans in the clinical two-hour, nine-minute performance that also included 10 aces.
He is looking to eclipse his previous best Open run - reaching the quarter-finals in 2021 - while he has never gone further than the last eight at any major.
Rublev was satisfied with his showing and felt his confidence was growing with each match.
He came into the Open in questionable form after losing both matches he played in the two Adelaide tournaments, losing to home hope Thanasi Kokkinakis and Spaniard Roberto Bautista Agut.
"Every match I'm playing better and better and I'm really happy because at the beginning of the season I lost a couple of tough matches against great players," Rublev said.
"I lost a bit of confidence and I'm happy that since the first match here I'm playing really good tennis and I'm improving."
Teen star Rune fell heavily, rolling his ankle as he chased down a shot, with a concerned Humbert crossing the net to check on his opponent.
Play was stopped on John Cain Arena for five minutes as the world No.10 had medical treatment, also hurting his wrist in the fall.
But Rune was able to shake it off to continue his best ever Open run.
"I was a bit worried - my ankle is good but my wrist was hurting a bit," the 19-year-old said.
"Of course it was painful but I tried to focus on something else - I'm playing on such an amazing stage and in the third round of a grand slam so I didn't think about it, I just kept going."
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