Second seed Dominic Thiem started the Vienna Open with a 6-4 6-3 defeat of Russian Andrey Rublev as the Austrian scored the 169th singles win of his career on Tuesday.
The world No.6, who has qualified for next month's ATP Finals in London, is hoping to improve on his home record of never advancing past the second round at the Stadthalle in five previous appearances.
The Austrian is the top box-office draw at an event in which Stockholm finalist Grigor Dimitrov withdrew due to fatigue.
Thiem led compatriot Dennis Novak into the second round as the youngster beat Italy's Thomas Fabbiano 7-6 (7-4) 7-5.
But local Sebastian Ofner fell 6-3 6-4 to Frenchman Lucas Pouille.
Thiem was not entirely satisfied with his performance against Rublev, a US Open quarter-finalist.
"He's a dangerous first-round opponent who goes for everything," Thiem said. "I didn't play my best match.
"But I'm once glad I get the chance to reach the next round. It always helps playing in your own country. It was very positive for me, and it was certainly not easy for Rublev."
Thiem is wary of Gasquet, who won their only previous meeting in Basel two years ago.
"He's completely different player than Rublev. He has more finesse and plays a great backhand," he said.
Fourth seed Pablo Carreno Busta beat Argentine Guido Pella 6-2 7-6 (9-7) while Spain's Albert Ramos-Vinolas put out US sixth seed Sam Querrey 3-6 7-6 (9-7) 7-6 (7-3).
German Jan-Lennard Struff beat Pablo Cuevas 7-5 6-1.
Spain's Guillermo Garcia-Lopez upset seventh seed Kevin Anderson 6-4 6-4 and Briton Kyle Edmund knocked out David Ferrer 6-2 7-6 (7-5).
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