Coric held his nerve to beat the world number 10 and will face Georgia's Nikoloz Basilashvili in the second round.
"It was such a tough match," Coric said after the game. "I think he was serving two times for the match and I was just thinking to myself (that) I was playing good for the whole match, and that was my main focus."
The Czech Republic's Berdych slipped early in his match and lost the first set before recovering to win the second in dominant fashion. But German Struff dug deep to force a tie-break in the final set and went on to win 6-4 1-6 7-6 (4).
Struff, who is ranked 53rd in the world, will take on the winner of the match between Italian Paolo Lorenzi and France's Gael Monfils in the next stage.
Aljaz Bedene, now representing his home country of Slovenia again, set up a second-round encounter with top seed Dominic Thiem after beating Tunisian wildcard Malek Jaziri 7-5 6-1.
Sixth seed Filip Krajinovic also lost to unseeded German Peter Gojowczyk, who will take on Italian Matteo Berrettini in the next round while eighth-seeded Spaniard Feliciano Lopez is set to face Bosnia's Mirza Basic.
Qatari Jabor Al-Mutawa was defeated by Italy's Stefano Travaglia, who fill face either fourth-seeded Albert Ramos-Vinolas or Argentine Guido Pella.
(Reporting by Aditi Prakash in Bengaluru,; Editing by Neville Dalton)
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