The world number four, looking for his third title in Chennai, showed little sympathy for his 18-year-old opponent and raced into a 5-0 lead in the first set.
World number 95 Coric did well to hold his serve in the sixth game and get on the board but could not stop the Swiss from taking the first set.
The second set went on serve till Wawrinka got the break in the seventh game before racing to victory.
Wawrinka won 24 out of 26 points on his first serve and such was his domination that Coric could not manage a single break point against him.
He will next play eighth-seeded Gilles Mueller of Luxembourg, who defeated Elias Ymer of Sweden 6-4 6-3, on Friday.
Fourth-seeded David Goffin of Belgium survived a scare against Lithuanian Ricardas Berankis to win 6-0 4-6 7-6 (1) and advance to the last eight.
(Writing by Sudipto Ganguly in Mumbai; editing by Ed Osmond)
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