Unseeded Evans, ranked 26th in the world, overcame his Russian opponent 6-2 6-2 in an hour and 16 minutes.
The 32-year-old will face Denis Shapovalov in the last eight after the Canadian edged past American Taylor Fritz 6-1 4-6 6-3.
Also on Wednesday, top seed Daniil Medvedev and second seed Stefanos Tsitsipas each progressed to the last 16.
Medvedev set up a meeting with 2019 champion and home favourite Dominic Thiem by beating Nikoloz Basilashvili 6-2 6-2, while Tsitsipas defeated Dennis Novak 7-6 (2) 6-2.
Elsewhere, Italy's Jannik Sinner overcame Chilean Cristian Garin 6-3 6-2.
British number one Cameron Norrie awaits Marcos Giron following the American's 6-4 6-2 success over Dutchman Tallon Griekspoor.
At the Swiss Indoors Basel tournament, world number one Carlos Alcaraz booked his place in the quarter-finals thanks to a 6-4 6-2 victory over Botic van de Zandschulp.
In Wednesday's first-round matches, third seed Felix Auger-Aliassime battled back to overcome Swiss player Marc-Andrea Huesler 6-7 (3) 6-4 6-4 but seventh seed Alex de Minaur was beaten 6-2 7-5 by Holger Rune, while eighth seed Lorenzo Musetti surrendered a one-set lead to lose 1-6 6-3 6-4 to Albert Ramos-Vinolas.
French qualifiers Ugo Humbert and Arthur Rinderknech progressed to the last 16 and last eight respectively. Humbert beat American Jenson Brooksby 6-1 6-4, with Rinderknech overcoming Slovakia's Alex Molcan 6-2 6-4.