The Argentine overcame the 34-year-old Frenchman 7-5, 6-3 to reach the third round at the last Masters 1000 tournament of the year.
Schwartzman, who will next face the Spanish qualifier Alejandro Davidovich Fokina, will play in the eight-man ATP Finals in London by reaching the semi-finals in Paris.
Should he be eliminated before the last four, he would qualify so long as Pablo Carreno Busta, Milos Raonic and Stan Wawrinka fail to lift the title on Sunday.
Destiny
"I have it in my hands,” said the 28-year-old after registering his first win over Gasquet at the fourth attempt. “I have to say that I'm watching the other matches. I want to be there in London so I'm doing my best on court.”
An appearance at the exclusive finale from 15 November at the 02 Arena would cap a brilliant year for Schwartzman.
He reached his first Grand Slam semi-final at last month's French Open in Paris a few weeks after reaching the final at the Rome Masters.
"Richard is such a good player,” added Schwartzman. “I’d never beaten him before so it was tough but I think I played a really good match.”
Third seed Daniil Medvedev also progressed to the third round after Kevin Anderson retired from their tie with a thigh injury during the tiebreak in the first set.
