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.
The 25-year-old will face the winner of the clash between Danish ninth seed Holger Rune and Frenchman Ugo Humbert for a place in the quarter-finals.
Mid-match Evans showed his class when he gave 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.
"After the first set I felt confidence and that I can play better and play faster and add one more speed and as soon as I did it I started to feel even better and I feel he's not comfortable.
"I tried to play even more aggressive and at the end played really good tennis."
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