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The Russian qualifier broke the third seeded Swiss in his first two service games and won 81 per cent of his first serve points to wrap up the upset in just over an hour.
The 21-year-old shed tears of joy after pulling off the biggest win of his career.
"It is such an amazing feeling when you're playing a legend like Roger and all these people supporting him until the end," he told ESPN in an on-court interview.
"One day I hope I'm going to feel the same.
"Today I was just trying to do my best and was thinking in my head that I need to play every point until the end no matter what."
The win sets up an all Russian quarter-final with ninth seed Daniil Medvedev.
Chris Froome (Stage 3, 2015)
James Toseland (1/2)