The three-time Grand Slam winner, playing in his first match since being defeated in a five-set epic by Marton Fucsovics at the Australian Open, was beaten by world number 21 Khachanov in 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.
The in-form Russian will face former world number one Andy Murray in the second round.
Second seed Stefanos Tsitsipas eased into round two with a 7-6 (7/4), 7-5 victory over Belarusian Egor Gerasimov.
The Greek, who beat Rafael Nadal en route to the Australian Open semi-finals last month, dropped his serve once in each set but did enough to set up a last-16 tie with Poland's Hubert Hurkacz, who saw off Frenchman Adrian Mannarino 6-3, 7-6 (8/6).