The Briton won the first set 7-6(3), clinching it on the fourth attempt and then broke the Russian in the eighth game of the second set.
Evans - who was banned for a year in 2017 for taking cocaine - has won 10 straight matches after also winning the title at Surbiton last week.
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The world No 70 - who will face Stan Wawrinka at Queen’s next week - is already certain to appear in the main draw at Wimbledon and could be the best British hope of getting to the second week.
Evans, 29, is behind Kyle Edmund and Cameron Norrie in the world rankings but on current form - and the patchy period his compatriots are going through in 2019 - he could progress through the first few rounds if the draw is kind to him.
“It’s been a good week, carrying on strongly from last week,” the 29-year-old said. “I played very well in the first match [today]. The second match, I was a bit frustrated and I was very tired.
"But I didn’t lose a set today, so I can’t be too unhappy. I played well out there, despite the wind.”
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