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The three-time Grand Slam winner, playing in his first ATP match since losing a five-set epic to Marton Fucsovics at the Australian Open, was put away by world number 21 Khachanov in a fairly comfortable fashion.
Big-hitter Khachanov eased through the match without dropping a service game to eighth seed Wawrinka.
The 24 year-old will face British qualifier Cameron Norrie in the last 16.
Khachanov's countryman and fourth seed Andrey Rublev strolled into the next round thanks to a 7-6 (8/1), 6-3 win over American qualifier Marcos Giron.
World number-six Stefanos Tsitsipas takes on Egor Gerasimov in his first match since falling at the semi-final stage of the Australian Open to Rotterdam top seed Daniil Medvedev.