It was Coric's first victory in four meetings against former world number three Wawrinka, who has struggled on his return from knee surgery this season and entered as a wildcard.
Coric, seeded 13th, will next face American qualifier Bradley Klahn, who claimed a hard-fought 4-6 6-4 6-3 win over Nick Kyrgios.
The 23-year-old Australian Kyrgios lost five straight games from 4-4 in the second set to hand the momentum to Klahn.
World number 104 Klahn. who fired down 18 aces, recorded just his fourth tour-level win of the season.
Greek youngster Stefanos Tsitsipas, whose run to the quarter-finals in Tokyo last week was enough to book a place at the ATP Finals in November, opened his Shanghai campaign with a 7-6(4) 4-6 6-4 win over Gael Monfils.
The 20-year-old broke the Frenchman's serve twice in the decider to set up a second round meeting with Russian Karen Khachanov.
"I have improved a lot since last year physically, mentally," said 10th seed Tsitsipas. "My game has changed. I am more aggressive now. I'm actually stronger. I feel stronger. My body has changed since last year.
"I have (now played) lots of matches in the ATP main draws, lots of wins which gave me confidence. I believe in myself a little bit more."
Canadian Milos Raonic suffered an early exit as the 2016 Wimbledon finalist lost to American qualifier Mackenzie McDonald 6-3 5-7 6-3.
Big-serving American Sam Querrey struck 14 aces on his way to a 6-3 6-4 win over world number 16 Diego Schwartzman and set up a second round meeting against his compatriot Taylor Fritz, who beat Robin Haase 6-3 7-6(3).
(Reporting by Hardik Vyas in Bengaluru; Editing by Ken Ferris)
