Yet world number 31 Gasquet, the fifth seed, was surprisingly beaten 6-3 6-4 in the late match by Greece's rising talent Stefanos Tsitsipas, with the 19-year-old from Athens progressing to face Thiem in the next round.
Seventh-seeded Fernando Verdasco was defeated 6-4 3-6 6-4 by Russian youngster Andrey Rublev while eighth seed Feliciano Lopez suffered a shock 6-4 7-6(2) loss to Bosnia's Mirza Basic, who is ranked 138th in the world.
The Spanish duo joined high-ranking players Pablo Carreno Busta, Tomas Berdych, Albert Ramos-Vinolas and Filip Krajinovic in exiting the tournament in the early stages.
In the quarter-finals, Rublev, 20, will face Croatia's Borna Coric, who ousted Carreno Busta in the first round, while Basic will take on Guido Pella after the Argentine overcame Stefano Travaglia 7-6(4) 6-3.
French wildcard Gael Monfils beat Jan-Lennard Struff 6-3 1-6 6-3 to set up the other last-eight clash with German Peter Gojowczyk.
(Reporting by Aditi Prakash in Bengaluru,; Editing by Ian Chadband)
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