The world number four took just 52 minutes to dispatch Russian wildcard Andrey Rublev in straight sets 6-3, 6-2 to set up a last-eight clash with Spaniard Guillermo Garcia-Lopez.
Fifth seed Garcia-Lopez had earlier defeated Croatian qualifier Ante Pavic 7-6 (7/5), 6-1 in their second-round tie at the Nungambakkam Tennis Stadium on Wednesday.
Two-time Grand Slam winner Wawrinka, who is the top seed in India, is seeking to get his name on the Chennai Open trophy for the fourth time after victories in 2011, 2014 and 2015.
And after receiving a first-round bye he got off to a emphatic start, hitting six aces and breaking his 18-year-old Russian opponent on three occasions in Wednesday's clash.
Elsewhere in the last 16 on Wednesday, number eight seed Borna Coric of Croatia defeated American Austin Krajicek 6-2, 7-5, while fourth seed Roberto Bautista Agut defeated John Millman 6-4, 6-1.
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The final of the Chennai Open takes place on Sunday.